diff --git a/docs/.vitepress/config.js b/docs/.vitepress/config.js index fb862f8..bf65b8c 100644 --- a/docs/.vitepress/config.js +++ b/docs/.vitepress/config.js @@ -207,7 +207,7 @@ function getOGTag(pageData) { ]); pageData.frontmatter.head.push([ "meta", - { property: "twitter:image", content: pageData.frontmatter.description }, + { property: "twitter:image", content: pageData.frontmatter.img }, ]); const metaData = { "@context": "http://schema.org", diff --git a/docs/.vitepress/dist/404.html b/docs/.vitepress/dist/404.html index b0bc58c..d0d8997 100644 --- a/docs/.vitepress/dist/404.html +++ b/docs/.vitepress/dist/404.html @@ -18,7 +18,7 @@
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- + \ No newline at end of file diff --git a/docs/.vitepress/dist/assets/contents_KG_ontology_ontology-chap-1.md.ByDWd7oq.js b/docs/.vitepress/dist/assets/contents_KG_ontology_ontology-chap-1.md.5lktpohY.js similarity index 99% rename from docs/.vitepress/dist/assets/contents_KG_ontology_ontology-chap-1.md.ByDWd7oq.js rename to docs/.vitepress/dist/assets/contents_KG_ontology_ontology-chap-1.md.5lktpohY.js index 5fb0e44..821708c 100644 --- a/docs/.vitepress/dist/assets/contents_KG_ontology_ontology-chap-1.md.ByDWd7oq.js +++ b/docs/.vitepress/dist/assets/contents_KG_ontology_ontology-chap-1.md.5lktpohY.js @@ -1 +1 @@ -import{_ as e,c as t,o,a5 as n}from"./chunks/framework.BuWuHeYF.js";const g=JSON.parse('{"title":"온톨로지의 설계 과정","description":"","frontmatter":{"head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"undefined\\",\\n \\"description\\":\\"undefined\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description"}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card"}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description"}],["meta",{"property":"twitter:image"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description"}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/KG/ontology/ontology-chap-1.md","filePath":"contents/KG/ontology/ontology-chap-1.md","lastUpdated":1711443986000}'),a={name:"contents/KG/ontology/ontology-chap-1.md"},r=n('

온톨로지의 설계 과정 ​

1. 요구분석과 기초적인 시각화 ​

온톨로지는 요구사항에 맞춰 도메인의 클래스, 인스턴스, 논리적 포함관계(substances), 관계(association)을 정의한다. 이들에 대해 나타내는 용어를 '단어(term)'이라고 하며, 보통 설계 초기에는 단어를 정의하고 이들의 실질적 관계를 엑셀 시트 등의 테이블 형태 문서에 정리한다.

단어에 대한 정의가 완료되면, 이들의 관계를 그래프 구조로 간단하게 시각화 한다. 아주 기본적이고 형식적이지 않은 방법으로는 마인드맵, ? 의 방법이 있다. 이러한 방법은 기본적인 구조화 이전, 의뢰자와 같이 온톨로지나 지식그래프, 프로그래밍 자체에 대한 기본적인 지식이 없는 사람들에게 온톨로지에 대해 설명하기에 용이하다.

또한, 본격적인 설계에 앞서 시각화를 통해 단어들간의 전반적인 관계 구조를 파악하고, 이를 기반으로 단어의 재정의나 추가적인 정의 등 본격적인 설계 이전에 단어와 관계를 점검할 수 있다.

2. lightweight ontology ​

경량화 온톨로지란 UML이나 ?? 와 같은, 전 분야의 프로그래밍(주로 객체지향)에서 사용되는 모델링 시각화 방식으로 온톨로지를 시각화하는 방법이다. 대표적으로 UML이 사용된다. UML은 클래스와 인스턴스(객체)를 사각형의 도형으로 표현하고, 도형 안에 클래스의 타입 등의 규칙, 속성 등을 기재한다.

논리적 포함관계는 삼각형으로 채워진 화살표로, 연관관계는 채워지지 않은 화살표로 표현한다.

'포함(substance)'와 '구성(hasComponent)'의 차이 구성요소는 특정 클래스/인스턴스의 속성으로 정의되는, 클래스는 인스턴스를 구성하는 다른 클래스나 인스턴스로, 상속관계와는 관련이 없다. 반면, 포함관계는 특정 클래스에서 파생/상속되는 클래스를 의미한다.

3. OWL ​

OWL은 W3C에서 제정한 표준 온톨로지 모델링 언어이다. UML로 어느정도 구조화된 형식으로 온톨로지 모형을 표현했다면, 이를 온톨로지 모델링 표준 어휘로 더 구체/구조화 된 형식으로 표현한다.

표준화된 어휘란, 즉, 모든 온톨로지 설계자들이 공통적으로 사용하는 어휘라는 것이다. 즉 온톨로지 모델링에 있어 일종의 파이썬이나 자바같은 언어라는 것이다. 아무래도 UML은 온톨로지 모델링 뿐만 아니라 프로그래밍 전반에 사용되는 모델링 언어이다 보니, 온톨로지에서 표준적으로 사용하는 어휘나 관계, 구조나 규칙을 제대로 표현하는데는 한계가 있다.

온톨로지는 OWL로 구조화 됨으로써 기계가 OWL로 작성된 온톨로지 구조를 판독할 수 있게 되고, 모든 온톨로지 설계자들이 어느 도메인의, 어떤 언어 사용자가 설계 했던 간에 이해할 수 있다.

OWL은 기본적으로 RDFS(RDF Schema)의 일종이다. 따라서 RDF 표현 형식을 따라 표현한다.

XML과 마크업 언어

XML은 데이터를 정의하는 규칙을 제공하는 마크업 언어이다. 마크업 언어란 태그 등을 이용해 문서나 데이터 구조를 명기하는 언어로, 프로그래밍 언어에 속하지는 않는다. 컴퓨터 등의 기계에 어떤 계산 작업을 수행하도록 지시하는 언어가 아니라, 단순히 문서/데이터 구조를 표현하는 언어이기 때문이다. XML은 <>를 이용해 누구나 자신만의 문서/데이터 구조를 표현할 수 있도록 한다.

다른 마크업 언어로는 HTML이 있다. HTML은 <h1>등의 태그를 활용해 웹 문서의 구조를 나타낸다.

RDF, RDFS

RDF란 웹 상의 정보를 표현하기 위한 규격이다. HTML이 웹 문서 내용을 구조화한다면, RDF는 웹 문서의 메타 정보를 구조화하여 나타내는 프레임워크이다. RDF는 각기 다른 도메인에서 정의되는 동의어를 의미를 분명하게 구분하기 위해 XML의 namespace를 이용한다. OWL에서는 그래프 구조를 표현하기 위해 RDF 형식을 차용한다.

RDFS는 특정 메타데이터에서 정의하고 있는 어휘들을 선언하기 위해 사용된다. 즉, 어떤 도메인에서 표준적으로 활용하기 위해 도메인에 적합하도록 사용어휘나 표현 규칙을 체계화하여 RDF를 표현형식대로 자료를 기술하는 어휘 체계를 의미한다. RDFS의 일종인 OWL은, 웹 자원에 대한 메타 정보를 온톨로지 형태로 표준화하여 기술하도록 하는 어휘 체계라고 할 수 있다.

',16),p=[r];function i(c,s,l,d,m,h){return o(),t("div",null,p)}const u=e(a,[["render",i]]);export{g as __pageData,u as default}; +import{_ as e,c as t,o,a5 as n}from"./chunks/framework.BuWuHeYF.js";const g=JSON.parse('{"title":"온톨로지의 설계 과정","description":"","frontmatter":{"head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"undefined\\",\\n \\"description\\":\\"undefined\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description"}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card"}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description"}],["meta",{"property":"twitter:image"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description"}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/KG/ontology/ontology-chap-1.md","filePath":"contents/KG/ontology/ontology-chap-1.md","lastUpdated":1712018592000}'),a={name:"contents/KG/ontology/ontology-chap-1.md"},r=n('

온톨로지의 설계 과정 ​

1. 요구분석과 기초적인 시각화 ​

온톨로지는 요구사항에 맞춰 도메인의 클래스, 인스턴스, 논리적 포함관계(substances), 관계(association)을 정의한다. 이들에 대해 나타내는 용어를 '단어(term)'이라고 하며, 보통 설계 초기에는 단어를 정의하고 이들의 실질적 관계를 엑셀 시트 등의 테이블 형태 문서에 정리한다.

단어에 대한 정의가 완료되면, 이들의 관계를 그래프 구조로 간단하게 시각화 한다. 아주 기본적이고 형식적이지 않은 방법으로는 마인드맵, ? 의 방법이 있다. 이러한 방법은 기본적인 구조화 이전, 의뢰자와 같이 온톨로지나 지식그래프, 프로그래밍 자체에 대한 기본적인 지식이 없는 사람들에게 온톨로지에 대해 설명하기에 용이하다.

또한, 본격적인 설계에 앞서 시각화를 통해 단어들간의 전반적인 관계 구조를 파악하고, 이를 기반으로 단어의 재정의나 추가적인 정의 등 본격적인 설계 이전에 단어와 관계를 점검할 수 있다.

2. lightweight ontology ​

경량화 온톨로지란 UML이나 ?? 와 같은, 전 분야의 프로그래밍(주로 객체지향)에서 사용되는 모델링 시각화 방식으로 온톨로지를 시각화하는 방법이다. 대표적으로 UML이 사용된다. UML은 클래스와 인스턴스(객체)를 사각형의 도형으로 표현하고, 도형 안에 클래스의 타입 등의 규칙, 속성 등을 기재한다.

논리적 포함관계는 삼각형으로 채워진 화살표로, 연관관계는 채워지지 않은 화살표로 표현한다.

'포함(substance)'와 '구성(hasComponent)'의 차이 구성요소는 특정 클래스/인스턴스의 속성으로 정의되는, 클래스는 인스턴스를 구성하는 다른 클래스나 인스턴스로, 상속관계와는 관련이 없다. 반면, 포함관계는 특정 클래스에서 파생/상속되는 클래스를 의미한다.

3. OWL ​

OWL은 W3C에서 제정한 표준 온톨로지 모델링 언어이다. UML로 어느정도 구조화된 형식으로 온톨로지 모형을 표현했다면, 이를 온톨로지 모델링 표준 어휘로 더 구체/구조화 된 형식으로 표현한다.

표준화된 어휘란, 즉, 모든 온톨로지 설계자들이 공통적으로 사용하는 어휘라는 것이다. 즉 온톨로지 모델링에 있어 일종의 파이썬이나 자바같은 언어라는 것이다. 아무래도 UML은 온톨로지 모델링 뿐만 아니라 프로그래밍 전반에 사용되는 모델링 언어이다 보니, 온톨로지에서 표준적으로 사용하는 어휘나 관계, 구조나 규칙을 제대로 표현하는데는 한계가 있다.

온톨로지는 OWL로 구조화 됨으로써 기계가 OWL로 작성된 온톨로지 구조를 판독할 수 있게 되고, 모든 온톨로지 설계자들이 어느 도메인의, 어떤 언어 사용자가 설계 했던 간에 이해할 수 있다.

OWL은 기본적으로 RDFS(RDF Schema)의 일종이다. 따라서 RDF 표현 형식을 따라 표현한다.

XML과 마크업 언어

XML은 데이터를 정의하는 규칙을 제공하는 마크업 언어이다. 마크업 언어란 태그 등을 이용해 문서나 데이터 구조를 명기하는 언어로, 프로그래밍 언어에 속하지는 않는다. 컴퓨터 등의 기계에 어떤 계산 작업을 수행하도록 지시하는 언어가 아니라, 단순히 문서/데이터 구조를 표현하는 언어이기 때문이다. XML은 <>를 이용해 누구나 자신만의 문서/데이터 구조를 표현할 수 있도록 한다.

다른 마크업 언어로는 HTML이 있다. HTML은 <h1>등의 태그를 활용해 웹 문서의 구조를 나타낸다.

RDF, RDFS

RDF란 웹 상의 정보를 표현하기 위한 규격이다. HTML이 웹 문서 내용을 구조화한다면, RDF는 웹 문서의 메타 정보를 구조화하여 나타내는 프레임워크이다. RDF는 각기 다른 도메인에서 정의되는 동의어를 의미를 분명하게 구분하기 위해 XML의 namespace를 이용한다. OWL에서는 그래프 구조를 표현하기 위해 RDF 형식을 차용한다.

RDFS는 특정 메타데이터에서 정의하고 있는 어휘들을 선언하기 위해 사용된다. 즉, 어떤 도메인에서 표준적으로 활용하기 위해 도메인에 적합하도록 사용어휘나 표현 규칙을 체계화하여 RDF를 표현형식대로 자료를 기술하는 어휘 체계를 의미한다. RDFS의 일종인 OWL은, 웹 자원에 대한 메타 정보를 온톨로지 형태로 표준화하여 기술하도록 하는 어휘 체계라고 할 수 있다.

',16),p=[r];function i(c,s,l,d,m,h){return o(),t("div",null,p)}const u=e(a,[["render",i]]);export{g as __pageData,u as default}; diff --git a/docs/.vitepress/dist/assets/contents_KG_ontology_ontology-chap-1.md.ByDWd7oq.lean.js b/docs/.vitepress/dist/assets/contents_KG_ontology_ontology-chap-1.md.5lktpohY.lean.js similarity index 97% rename from docs/.vitepress/dist/assets/contents_KG_ontology_ontology-chap-1.md.ByDWd7oq.lean.js rename to docs/.vitepress/dist/assets/contents_KG_ontology_ontology-chap-1.md.5lktpohY.lean.js index fb23040..806838e 100644 --- a/docs/.vitepress/dist/assets/contents_KG_ontology_ontology-chap-1.md.ByDWd7oq.lean.js +++ b/docs/.vitepress/dist/assets/contents_KG_ontology_ontology-chap-1.md.5lktpohY.lean.js @@ -1 +1 @@ -import{_ as e,c as t,o,a5 as n}from"./chunks/framework.BuWuHeYF.js";const g=JSON.parse('{"title":"온톨로지의 설계 과정","description":"","frontmatter":{"head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"undefined\\",\\n \\"description\\":\\"undefined\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description"}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card"}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description"}],["meta",{"property":"twitter:image"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description"}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/KG/ontology/ontology-chap-1.md","filePath":"contents/KG/ontology/ontology-chap-1.md","lastUpdated":1711443986000}'),a={name:"contents/KG/ontology/ontology-chap-1.md"},r=n("",16),p=[r];function i(c,s,l,d,m,h){return o(),t("div",null,p)}const u=e(a,[["render",i]]);export{g as __pageData,u as default}; +import{_ as e,c as t,o,a5 as n}from"./chunks/framework.BuWuHeYF.js";const g=JSON.parse('{"title":"온톨로지의 설계 과정","description":"","frontmatter":{"head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"undefined\\",\\n \\"description\\":\\"undefined\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description"}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card"}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description"}],["meta",{"property":"twitter:image"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description"}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/KG/ontology/ontology-chap-1.md","filePath":"contents/KG/ontology/ontology-chap-1.md","lastUpdated":1712018592000}'),a={name:"contents/KG/ontology/ontology-chap-1.md"},r=n("",16),p=[r];function i(c,s,l,d,m,h){return o(),t("div",null,p)}const u=e(a,[["render",i]]);export{g as __pageData,u as default}; diff --git a/docs/.vitepress/dist/assets/contents_KG_protege_class-restriction.md.Db1h9334.js b/docs/.vitepress/dist/assets/contents_KG_protege_class-restriction.md.DlEHAMyQ.js similarity index 56% rename from docs/.vitepress/dist/assets/contents_KG_protege_class-restriction.md.Db1h9334.js rename to docs/.vitepress/dist/assets/contents_KG_protege_class-restriction.md.DlEHAMyQ.js index cf502f1..8394b48 100644 --- a/docs/.vitepress/dist/assets/contents_KG_protege_class-restriction.md.Db1h9334.js +++ b/docs/.vitepress/dist/assets/contents_KG_protege_class-restriction.md.DlEHAMyQ.js @@ -1 +1 @@ -import{_ as t,c as e,o as n,a5 as i}from"./chunks/framework.BuWuHeYF.js";const _=JSON.parse('{"title":"Restriction","description":"","frontmatter":{"head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"undefined\\",\\n \\"description\\":\\"undefined\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description"}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card"}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description"}],["meta",{"property":"twitter:image"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description"}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/KG/protege/class-restriction.md","filePath":"contents/KG/protege/class-restriction.md","lastUpdated":null}'),r={name:"contents/KG/protege/class-restriction.md"},a=i('

Restriction ​

1. Quantifier Restriction (OWL) ​

2. "hasValue" Restriction (OWL) ​

3 Cardinality Restriction (OWL) ​

4. Existential Restriction ​

5. Universal Restriction ​

',6),o=[a];function s(c,p,l,d,h,u){return n(),e("div",null,o)}const y=t(r,[["render",s]]);export{_ as __pageData,y as default}; +import{_ as t,c as e,o as n,a5 as i}from"./chunks/framework.BuWuHeYF.js";const _=JSON.parse('{"title":"Restriction","description":"","frontmatter":{"head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"undefined\\",\\n \\"description\\":\\"undefined\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description"}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card"}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description"}],["meta",{"property":"twitter:image"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description"}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/KG/protege/class-restriction.md","filePath":"contents/KG/protege/class-restriction.md","lastUpdated":1712018592000}'),r={name:"contents/KG/protege/class-restriction.md"},a=i('

Restriction ​

1. Quantifier Restriction (OWL) ​

2. "hasValue" Restriction (OWL) ​

3 Cardinality Restriction (OWL) ​

4. Existential Restriction ​

5. Universal Restriction ​

',6),o=[a];function s(c,p,l,d,h,u){return n(),e("div",null,o)}const y=t(r,[["render",s]]);export{_ as __pageData,y as default}; diff --git a/docs/.vitepress/dist/assets/contents_KG_protege_class-restriction.md.Db1h9334.lean.js b/docs/.vitepress/dist/assets/contents_KG_protege_class-restriction.md.DlEHAMyQ.lean.js similarity index 88% rename from docs/.vitepress/dist/assets/contents_KG_protege_class-restriction.md.Db1h9334.lean.js rename to docs/.vitepress/dist/assets/contents_KG_protege_class-restriction.md.DlEHAMyQ.lean.js index f647323..96c1da9 100644 --- a/docs/.vitepress/dist/assets/contents_KG_protege_class-restriction.md.Db1h9334.lean.js +++ b/docs/.vitepress/dist/assets/contents_KG_protege_class-restriction.md.DlEHAMyQ.lean.js @@ -1 +1 @@ -import{_ as t,c as e,o as n,a5 as i}from"./chunks/framework.BuWuHeYF.js";const _=JSON.parse('{"title":"Restriction","description":"","frontmatter":{"head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"undefined\\",\\n \\"description\\":\\"undefined\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description"}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card"}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description"}],["meta",{"property":"twitter:image"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description"}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/KG/protege/class-restriction.md","filePath":"contents/KG/protege/class-restriction.md","lastUpdated":null}'),r={name:"contents/KG/protege/class-restriction.md"},a=i("",6),o=[a];function s(c,p,l,d,h,u){return n(),e("div",null,o)}const y=t(r,[["render",s]]);export{_ as __pageData,y as default}; +import{_ as t,c as e,o as n,a5 as i}from"./chunks/framework.BuWuHeYF.js";const _=JSON.parse('{"title":"Restriction","description":"","frontmatter":{"head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"undefined\\",\\n \\"description\\":\\"undefined\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description"}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card"}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description"}],["meta",{"property":"twitter:image"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description"}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/KG/protege/class-restriction.md","filePath":"contents/KG/protege/class-restriction.md","lastUpdated":1712018592000}'),r={name:"contents/KG/protege/class-restriction.md"},a=i("",6),o=[a];function s(c,p,l,d,h,u){return n(),e("div",null,o)}const y=t(r,[["render",s]]);export{_ as __pageData,y as default}; diff --git a/docs/.vitepress/dist/assets/contents_LLM_finetuning_README.md.DonYs2WF.js b/docs/.vitepress/dist/assets/contents_LLM_finetuning_README.md.DonYs2WF.js new file mode 100644 index 0000000..e2f379f --- /dev/null +++ b/docs/.vitepress/dist/assets/contents_LLM_finetuning_README.md.DonYs2WF.js @@ -0,0 +1 @@ +import{_ as e,c as t,o as n,a5 as o}from"./chunks/framework.BuWuHeYF.js";const h=JSON.parse('{"title":"FineTuning 실습코드","description":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다.","frontmatter":{"description":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다.","head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다.\\",\\n \\"description\\":\\"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다.\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"twitter:image"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/LLM/finetuning/README.md","filePath":"contents/LLM/finetuning/README.md","lastUpdated":1711265401000}'),r={name:"contents/LLM/finetuning/README.md"},a=o('

FineTuning 실습코드 ​

깃허브 바로가기

목차 ​

아래 링크를 클릭하면 colab 노트북으로 넘어간다. 혹은 좌측 내비게이션 항목을 클릭해도 된다.

  1. GPT Finetuning
  2. Korquad 데이터셋으로 GPT 파인튜닝 해보기
',6),i=[a];function p(c,s,l,d,u,m){return n(),t("div",null,i)}const y=e(r,[["render",p]]);export{h as __pageData,y as default}; diff --git a/docs/.vitepress/dist/assets/contents_LLM_finetuning_README.md.mBZV9Uj4.lean.js b/docs/.vitepress/dist/assets/contents_LLM_finetuning_README.md.DonYs2WF.lean.js similarity index 62% rename from docs/.vitepress/dist/assets/contents_LLM_finetuning_README.md.mBZV9Uj4.lean.js rename to docs/.vitepress/dist/assets/contents_LLM_finetuning_README.md.DonYs2WF.lean.js index e484a7f..42db807 100644 --- a/docs/.vitepress/dist/assets/contents_LLM_finetuning_README.md.mBZV9Uj4.lean.js +++ b/docs/.vitepress/dist/assets/contents_LLM_finetuning_README.md.DonYs2WF.lean.js @@ -1 +1 @@ -import{_ as e,c as t,o as n,a5 as o}from"./chunks/framework.BuWuHeYF.js";const h=JSON.parse('{"title":"FineTuning 실습코드","description":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다.","frontmatter":{"description":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다.","head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다.\\",\\n \\"description\\":\\"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다.\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"twitter:image","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/LLM/finetuning/README.md","filePath":"contents/LLM/finetuning/README.md","lastUpdated":1711265401000}'),r={name:"contents/LLM/finetuning/README.md"},a=o("",6),i=[a];function p(c,s,l,d,u,m){return n(),t("div",null,i)}const y=e(r,[["render",p]]);export{h as __pageData,y as default}; 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diff --git a/docs/.vitepress/dist/assets/contents_LLM_finetuning_README.md.mBZV9Uj4.js b/docs/.vitepress/dist/assets/contents_LLM_finetuning_README.md.mBZV9Uj4.js deleted file mode 100644 index 5dca787..0000000 --- a/docs/.vitepress/dist/assets/contents_LLM_finetuning_README.md.mBZV9Uj4.js +++ /dev/null @@ -1 +0,0 @@ -import{_ as e,c as t,o as n,a5 as o}from"./chunks/framework.BuWuHeYF.js";const h=JSON.parse('{"title":"FineTuning 실습코드","description":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다.","frontmatter":{"description":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다.","head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다.\\",\\n \\"description\\":\\"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다.\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"twitter:image","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description","content":"전체 코드와 자료는 아래 깃허브에 모아놓았으니 필요시 다운받아 사용하면 된다."}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/LLM/finetuning/README.md","filePath":"contents/LLM/finetuning/README.md","lastUpdated":1711265401000}'),r={name:"contents/LLM/finetuning/README.md"},a=o('

FineTuning 실습코드 ​

깃허브 바로가기

목차 ​

아래 링크를 클릭하면 colab 노트북으로 넘어간다. 혹은 좌측 내비게이션 항목을 클릭해도 된다.

  1. GPT Finetuning
  2. Korquad 데이터셋으로 GPT 파인튜닝 해보기
',6),i=[a];function p(c,s,l,d,u,m){return n(),t("div",null,i)}const y=e(r,[["render",p]]);export{h as __pageData,y as default}; diff --git a/docs/.vitepress/dist/assets/contents_MATH_automatic-differentiate.md.CVh05dT0.js b/docs/.vitepress/dist/assets/contents_MATH_automatic-differentiate.md.CVh05dT0.js deleted file mode 100644 index 9ef14b3..0000000 --- a/docs/.vitepress/dist/assets/contents_MATH_automatic-differentiate.md.CVh05dT0.js +++ /dev/null @@ -1 +0,0 @@ -import{_ as e,c as n,o,m as t,a}from"./chunks/framework.BuWuHeYF.js";const D=JSON.parse('{"title":"🦾 Automatic Differentiation (자동미분)","description":"A.K.A AutoDiff, AutoGrad","frontmatter":{"description":"A.K.A AutoDiff, AutoGrad","head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"A.K.A AutoDiff, AutoGrad\\",\\n \\"description\\":\\"A.K.A AutoDiff, AutoGrad\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"twitter:image","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/MATH/automatic-differentiate.md","filePath":"contents/MATH/automatic-differentiate.md","lastUpdated":1711265401000}'),i={name:"contents/MATH/automatic-differentiate.md"},r=t("h1",{id:"🦾-automatic-differentiation-자동미분",tabindex:"-1"},[a("🦾 Automatic Differentiation (자동미분) "),t("a",{class:"header-anchor",href:"#🦾-automatic-differentiation-자동미분","aria-label":'Permalink to "🦾 Automatic Differentiation (자동미분)"'},"​")],-1),p=t("p",null,"자동미분은 델타논법과 같은 수치 미줌과는 구분되는 미분 방식이다. 수치 미분의 경우 기호위주의 대수학적 규칙을 컴퓨터 연산에 적용하다 보니 비효율적이며, 반올림 에러가 발생한다는 문제가 있다.",-1),c=t("p",null,"따라서, 다수의 input에 대한 미분값을 구해야 하는 컴퓨터 연산 환경에서는 자동미분을 활용한다. 자동미분은 기본적으로 편미분의 연쇄법칙과 같은 연쇄법칙을 적용하여 계산하며, y->x의 순서로 미분 연산을 진행한다. ",-1),d=t("p",null,"자동미분은 딥러닝과 머신러닝에서 아주 기본적이고 핵심적인 연산으로, 역전파 방식을 통한 가중치 갱신 과정에 활용된다.",-1),s=t("p",null,"PyTorch와 Tensorflow는 자동미분을 계산하는 라이브러리이며, 이들을 활용해 딥러닝의 신경망을 구현한다. 이 두 라이브러리에서 자동미분은 역전파(backward pass)로 계산된다.",-1),u=[r,p,c,d,s];function f(m,A,l,y,h,_){return o(),n("div",null,u)}const b=e(i,[["render",f]]);export{D as __pageData,b as default}; diff --git a/docs/.vitepress/dist/assets/contents_MATH_automatic-differentiate.md.CVh05dT0.lean.js b/docs/.vitepress/dist/assets/contents_MATH_automatic-differentiate.md.CVh05dT0.lean.js deleted file mode 100644 index 9ef14b3..0000000 --- a/docs/.vitepress/dist/assets/contents_MATH_automatic-differentiate.md.CVh05dT0.lean.js +++ /dev/null @@ -1 +0,0 @@ -import{_ as e,c as n,o,m as t,a}from"./chunks/framework.BuWuHeYF.js";const D=JSON.parse('{"title":"🦾 Automatic Differentiation (자동미분)","description":"A.K.A AutoDiff, AutoGrad","frontmatter":{"description":"A.K.A AutoDiff, AutoGrad","head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"A.K.A AutoDiff, AutoGrad\\",\\n \\"description\\":\\"A.K.A AutoDiff, AutoGrad\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"twitter:image","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/MATH/automatic-differentiate.md","filePath":"contents/MATH/automatic-differentiate.md","lastUpdated":1711265401000}'),i={name:"contents/MATH/automatic-differentiate.md"},r=t("h1",{id:"🦾-automatic-differentiation-자동미분",tabindex:"-1"},[a("🦾 Automatic Differentiation (자동미분) "),t("a",{class:"header-anchor",href:"#🦾-automatic-differentiation-자동미분","aria-label":'Permalink to "🦾 Automatic Differentiation (자동미분)"'},"​")],-1),p=t("p",null,"자동미분은 델타논법과 같은 수치 미줌과는 구분되는 미분 방식이다. 수치 미분의 경우 기호위주의 대수학적 규칙을 컴퓨터 연산에 적용하다 보니 비효율적이며, 반올림 에러가 발생한다는 문제가 있다.",-1),c=t("p",null,"따라서, 다수의 input에 대한 미분값을 구해야 하는 컴퓨터 연산 환경에서는 자동미분을 활용한다. 자동미분은 기본적으로 편미분의 연쇄법칙과 같은 연쇄법칙을 적용하여 계산하며, y->x의 순서로 미분 연산을 진행한다. ",-1),d=t("p",null,"자동미분은 딥러닝과 머신러닝에서 아주 기본적이고 핵심적인 연산으로, 역전파 방식을 통한 가중치 갱신 과정에 활용된다.",-1),s=t("p",null,"PyTorch와 Tensorflow는 자동미분을 계산하는 라이브러리이며, 이들을 활용해 딥러닝의 신경망을 구현한다. 이 두 라이브러리에서 자동미분은 역전파(backward pass)로 계산된다.",-1),u=[r,p,c,d,s];function f(m,A,l,y,h,_){return o(),n("div",null,u)}const b=e(i,[["render",f]]);export{D as __pageData,b as default}; diff --git a/docs/.vitepress/dist/assets/contents_MATH_automatic-differentiate.md.gEnpd4Ux.js b/docs/.vitepress/dist/assets/contents_MATH_automatic-differentiate.md.gEnpd4Ux.js new file mode 100644 index 0000000..6c54b69 --- /dev/null +++ b/docs/.vitepress/dist/assets/contents_MATH_automatic-differentiate.md.gEnpd4Ux.js @@ -0,0 +1 @@ +import{_ as e,c as n,o,m as t,a}from"./chunks/framework.BuWuHeYF.js";const D=JSON.parse('{"title":"🦾 Automatic Differentiation (자동미분)","description":"A.K.A AutoDiff, AutoGrad","frontmatter":{"description":"A.K.A AutoDiff, AutoGrad","head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsundefined\\"\\n },\\n \\"name\\":\\"undefined\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/undefined\\",\\n \\"headline\\":\\"A.K.A AutoDiff, AutoGrad\\",\\n \\"description\\":\\"A.K.A AutoDiff, AutoGrad\\",\\n \\"keywords\\" : [undefined],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title"}],["meta",{"property":"og:description","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"og:url"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"twitter:title"}],["meta",{"property":"twitter:description","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"twitter:image"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/undefined"}],["meta",{"property":"description","content":"A.K.A AutoDiff, AutoGrad"}],["meta",{"property":"keywords"}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/MATH/automatic-differentiate.md","filePath":"contents/MATH/automatic-differentiate.md","lastUpdated":1711265401000}'),i={name:"contents/MATH/automatic-differentiate.md"},r=t("h1",{id:"🦾-automatic-differentiation-자동미분",tabindex:"-1"},[a("🦾 Automatic Differentiation (자동미분) "),t("a",{class:"header-anchor",href:"#🦾-automatic-differentiation-자동미분","aria-label":'Permalink to "🦾 Automatic Differentiation (자동미분)"'},"​")],-1),p=t("p",null,"자동미분은 델타논법과 같은 수치 미줌과는 구분되는 미분 방식이다. 수치 미분의 경우 기호위주의 대수학적 규칙을 컴퓨터 연산에 적용하다 보니 비효율적이며, 반올림 에러가 발생한다는 문제가 있다.",-1),c=t("p",null,"따라서, 다수의 input에 대한 미분값을 구해야 하는 컴퓨터 연산 환경에서는 자동미분을 활용한다. 자동미분은 기본적으로 편미분의 연쇄법칙과 같은 연쇄법칙을 적용하여 계산하며, y->x의 순서로 미분 연산을 진행한다. 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1. 미분 ​

미분의 개념 ​

미분은 무수한 점들로 구성된 곡선의 순간 기울기, 즉 곡선의 어느 한 점에서의 기울기를 구하는 것이다. 미분에 대한 표기 방법은 다음과 같다.

1. 일반적 표기 방법 ​

',4),n={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},r={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.926ex",height:"2.396ex",role:"img",focusable:"false",viewBox:"0 -809 2177.5 1059","aria-hidden":"true"},d=e('',1),i=[d],m=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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method)"'},"​")],-1),w={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.179ex",height:"1.645ex",role:"img",focusable:"false",viewBox:"0 -716 1405 727","aria-hidden":"true"},f=e('',1),L=[f],k=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"x")])],-1),M={class:"info 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같다.",-1),S={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},j={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-1.767ex"},xmlns:"http://www.w3.org/2000/svg",width:"32.071ex",height:"5.07ex",role:"img",focusable:"false",viewBox:"0 -1460 14175.3 2241.1","aria-hidden":"true"},C=e('',1),q=[C],I=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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미분법칙 ​

1. 상수법칙 ​

상수에 대한 미분값은 0이다.

2. 제곱법칙 ​

제곱의 미분은 제곱되는 값을 상수로 곱하고, 제곱수를 -1 해준다.

3. 상수곱법칙 ​

제곱법칙에서, 제곱되는 값은 변수의 상수곱에 함께 곱해진다.

4. 덧셈법칙 ​

미분시에도 덧셈의 성질은 그대로 유지된다.

5. 곱셈법칙 ​

덧셈과 마찬가지로 곱셈의 성질도 유지된다.

6. 연쇄법칙 ​

',12),B={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},N={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"7.138ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 3155 1000","aria-hidden":"true"},E=e('',1),J=[E],O=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"f"),t("mo",{stretchy:"false"},"("),t("mi",null,"g"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",{stretchy:"false"},")"),t("mo",{stretchy:"false"},")")])],-1);function z(G,$,W,X,K,U){return T(),Q("div",null,[s,t("mjx-container",n,[(T(),Q("svg",r,i)),m]),h,c,t("mjx-container",p,[(T(),Q("svg",g,u)),x]),H,t("p",null,[a('미분이 "순간적인 기울기"라고 정의했다.즉, 순간이 얼마나 짧은 시간이 되었건 다음 시점의 속도와 현재 시점 속도의 차이값을 구해야 변화율을 구할 수 있는데, 이 찰나의 시간차를 표현한기 위해서 0으로 수렴하는 값을 더해 다음 순간을 정의한다. 여기서 0에 수렴하는 값을 '),t("mjx-container",w,[(T(),Q("svg",y,L)),k]),a(" 라고 한다. 즉, 델타논법이란 델타값을 사용해 한 지점의 다음 순간을 정의하고, 다음 순간에서의 값과 현재 값의 차이 값을 두 순간의 차인 델타값으로 나누어 순간 기울기를 구하는 방법이다.")]),t("div",M,[b,t("p",null,[a("여기서 극한의 개념이 적용되는데, 극한이란 쉽게 말해 어떤 값에 매우 근사하고 있는 값은 사실상 그 값과 다름이 없다는 수학적 약속이다. 그러니까 6에 매우 근사하는 값인 5.999999999를 사실상 6으로 보고 계산하여도 6으로 계산하였을 때의 값과 크게 차이나지 않으니, 이 5.999999를 6으로 두고 계산할 수 있다. 다만, 엄밀히 두 값은 다른 값이니 "),t("mjx-container",V,[(T(),Q("svg",v,D)),A]),a(" 기호를 써서 6에 근사하는 값임을 나타내야 한다.")])]),P,t("mjx-container",S,[(T(),Q("svg",j,q)),I]),R,t("p",null,[a("가장 중요한 법칙으로, 머신러닝에서 중요한 법칙이다. 변수값으로 함수값이 들어간 복합함수의 경우 (e.g. "),t("mjx-container",B,[(T(),Q("svg",N,J)),O]),a(") 적용되는 법칙이다. 복합함수를 임의의 변수로 치환하여, 해당 변수에 대해 미분한 다음, 변수에 중첩된 함수를 해당 함수의 변수값으로 미분한 값을, 치환값의 미분값에 곱해준다.")])])}const t1=o(l,[["render",z]]);export{Y as __pageData,t1 as default}; diff --git a/docs/.vitepress/dist/assets/contents_MATH_calculus_cal-chap-1.md.BLV24G7b.lean.js b/docs/.vitepress/dist/assets/contents_MATH_calculus_cal-chap-1.md.BLV24G7b.lean.js deleted file mode 100644 index cd0f698..0000000 --- a/docs/.vitepress/dist/assets/contents_MATH_calculus_cal-chap-1.md.BLV24G7b.lean.js +++ /dev/null @@ -1 +0,0 @@ -import{_ as o,c as Q,m as t,a,a5 as e,o as T}from"./chunks/framework.BuWuHeYF.js";const Y=JSON.parse('{"title":"1. 미분","description":"미분의 개념과 미분법칙","frontmatter":{"title":"1. 미분","description":"미분의 개념과 미분법칙","keywords":["미분","미분법칙"],"url":"MATH/calculus/cal-chap-1.html","head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : 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'),t("mjx-container",w,[(T(),Q("svg",y,L)),k]),a(" 라고 한다. 즉, 델타논법이란 델타값을 사용해 한 지점의 다음 순간을 정의하고, 다음 순간에서의 값과 현재 값의 차이 값을 두 순간의 차인 델타값으로 나누어 순간 기울기를 구하는 방법이다.")]),t("div",M,[b,t("p",null,[a("여기서 극한의 개념이 적용되는데, 극한이란 쉽게 말해 어떤 값에 매우 근사하고 있는 값은 사실상 그 값과 다름이 없다는 수학적 약속이다. 그러니까 6에 매우 근사하는 값인 5.999999999를 사실상 6으로 보고 계산하여도 6으로 계산하였을 때의 값과 크게 차이나지 않으니, 이 5.999999를 6으로 두고 계산할 수 있다. 다만, 엄밀히 두 값은 다른 값이니 "),t("mjx-container",V,[(T(),Q("svg",v,D)),A]),a(" 기호를 써서 6에 근사하는 값임을 나타내야 한다.")])]),P,t("mjx-container",S,[(T(),Q("svg",j,q)),I]),R,t("p",null,[a("가장 중요한 법칙으로, 머신러닝에서 중요한 법칙이다. 변수값으로 함수값이 들어간 복합함수의 경우 (e.g. "),t("mjx-container",B,[(T(),Q("svg",N,J)),O]),a(") 적용되는 법칙이다. 복합함수를 임의의 변수로 치환하여, 해당 변수에 대해 미분한 다음, 변수에 중첩된 함수를 해당 함수의 변수값으로 미분한 값을, 치환값의 미분값에 곱해준다.")])])}const t1=o(l,[["render",z]]);export{Y as __pageData,t1 as default}; diff --git a/docs/.vitepress/dist/assets/contents_MATH_calculus_cal-chap-1.md.CGALZlOM.js b/docs/.vitepress/dist/assets/contents_MATH_calculus_cal-chap-1.md.CGALZlOM.js new 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개념과 미분법칙"}],["meta",{"property":"keywords","content":["미분","미분법칙"]}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/MATH/calculus/cal-chap-1.md","filePath":"contents/MATH/calculus/cal-chap-1.md","lastUpdated":1712018592000}'),l={name:"contents/MATH/calculus/cal-chap-1.md"},s=e('

1. 미분 ​

미분의 개념 ​

미분은 무수한 점들로 구성된 곡선의 순간 기울기, 즉 곡선의 어느 한 점에서의 기울기를 구하는 것이다. 미분에 대한 표기 방법은 다음과 같다.

1. 일반적 표기 방법 ​

',4),n={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},r={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.926ex",height:"2.396ex",role:"img",focusable:"false",viewBox:"0 -809 2177.5 1059","aria-hidden":"true"},d=e('',1),i=[d],m=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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2067","aria-hidden":"true"},_=e('',1),u=[_],x=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mfrac",null,[t("mi",null,"d"),t("mrow",null,[t("mi",null,"d"),t("mi",null,"x")])]),t("mi",null,"f"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",{stretchy:"false"},")")])],-1),H=t("h2",{id:"델타논법-delta-method",tabindex:"-1"},[a("델타논법 (delta method) "),t("a",{class:"header-anchor",href:"#델타논법-delta-method","aria-label":'Permalink to "델타논법 (delta method)"'},"​")],-1),w={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},y={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"3.179ex",height:"1.645ex",role:"img",focusable:"false",viewBox:"0 -716 1405 727","aria-hidden":"true"},f=e('',1),L=[f],k=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",{mathvariant:"normal"},"Δ"),t("mi",null,"x")])],-1),M={class:"info 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같다.",-1),S={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},j={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-1.767ex"},xmlns:"http://www.w3.org/2000/svg",width:"32.071ex",height:"5.07ex",role:"img",focusable:"false",viewBox:"0 -1460 14175.3 2241.1","aria-hidden":"true"},C=e('',1),q=[C],I=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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미분법칙 ​

1. 상수법칙 ​

상수에 대한 미분값은 0이다.

2. 제곱법칙 ​

제곱의 미분은 제곱되는 값을 상수로 곱하고, 제곱수를 -1 해준다.

3. 상수곱법칙 ​

제곱법칙에서, 제곱되는 값은 변수의 상수곱에 함께 곱해진다.

4. 덧셈법칙 ​

미분시에도 덧셈의 성질은 그대로 유지된다.

5. 곱셈법칙 ​

덧셈과 마찬가지로 곱셈의 성질도 유지된다.

6. 연쇄법칙 ​

',12),B={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},N={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"7.138ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 3155 1000","aria-hidden":"true"},E=e('',1),J=[E],O=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"f"),t("mo",{stretchy:"false"},"("),t("mi",null,"g"),t("mo",{stretchy:"false"},"("),t("mi",null,"x"),t("mo",{stretchy:"false"},")"),t("mo",{stretchy:"false"},")")])],-1);function z(G,$,W,X,K,U){return 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Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/MATH/linear-algebra-application/intermediate-chap-1.md","filePath":"contents/MATH/linear-algebra-application/intermediate-chap-1.md","lastUpdated":1711331516000}'),s={name:"contents/MATH/linear-algebra-application/intermediate-chap-1.md"},o=T('

1. 고유값과 고유벡터 ​

고유벡터와 고유값 ​

고유하다는 것은 상황이 변화해도 그 특성을 잃지 않는 것을 의미한다. 그럼, 벡터가 고유하다는 것은 무엇일까? 벡터가 어떠한 상황에서도 그 특성, 즉 방향성을 잃지 않는 것을 의미한다. 즉, 고유벡터란 선형변환 이후에도 변환 결과가 자신의 상수배를 한 결과일 때의 벡터를 의미한다. 선형 변환이란 쉽게 말해 어떤 행렬을 벡터에 곱하는 것이다.

여러가지 선형변환들

선형변환이란?

벡터에 어떠한 행렬을 곱하는 것을 '벡터에 행렬을 통과시킨다'라고 표현한다. 아무튼, 벡터에 어떠한 행렬을 곱하게 되면 벡터의 크기와 방향이 변한다.

하지만, 아무리 벡터의 크기나 방향이 변해 봤자, 조금 더 벡터의 크기가 커진다거나, 벡터의 방향이 치우치는 정도에 그치게 된다. 벡터가 곡선의 형상을 띄거나 하지는 않는다는 것이다. 이를 '선형적으로 변화했다'라고 말한다.

이처럼, 벡터가 어떠한 행렬을 통과하여 선형적인 변화를 일으키게 하는 것을 선형변환 (linear transformation) 이라고 한다.

선형변환에 속하는 다양한 변환들이 있으며, 모든 변환들은 행렬을 곱하여 이뤄진다. 어찌보면, 벡터에 곱해지는 행렬이 곧 함수와도 같은 역할을 한다고 볼 수 있다. 이런 변환들은 주로 컴퓨터 그래픽에 많이 활용된다. 하지만 아핀변환이라는 것은 LLM에서도 자주 언급되긴 한다.

▶️scailing (비례변환) 비례변환은 벡터의 방향과 크기가 변화하는 변환을 의미한다.

▶️Rotation (회전변환) 회전변환은 좌표평면이 원점을 중심으로 회전하는 것을 의미한다. 회전변환된 벡터는 원래의 벡터와 선형독립이며, 회전변환시 고유값과 고유벡터는 존재하지 않는다.

▶️Shearing (전단변환) 전단변환은 특정 차원값에만 변화를 주는 변환을 의미한다. 기하학적으로 이해하면, y축을 고정하고 x축 방향으로만 변화를 가하는 것을 의미한다.

앞서 살펴본 변환들은 모두 원점이 변화하지 않는 변환이다. 원점을 이동시키는 변환도 있다. 바로 이동변환 이라는 것인데, 대표적으로 아핀변환(Affine) 이 있다.

아핀변환 (Affine Transformation) TBD

식으로 나타내면 다음과 같이 표현할 수 있다.

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")],-1);function w7(u7,x7,_7,L7,f7,y7){return a(),Q("div",null,[o,t("p",null,[t("mjx-container",n,[(a(),Q("svg",d,r)),i])]),t("p",null,[e("여기서 "),t("mjx-container",h,[(a(),Q("svg",p,g)),H]),e(" 는 벡터에 곱해지는 스칼라를 의미한다. 이 스칼라 값인 람다를 "),w,e("이라고 칭한다. 즉, 고유값이란 고유벡터에 곱해지는 상수값을 의미한다.")]),u,t("p",null,[t("mjx-container",x,[(a(),Q("svg",_,f)),y])]),t("p",null,[t("mjx-container",k,[(a(),Q("svg",M,Z)),v])]),t("p",null,[t("mjx-container",b,[(a(),Q("svg",D,C)),A])]),t("p",null,[t("mjx-container",S,[(a(),Q("svg",R,I)),B])]),t("p",null,[t("mjx-container",O,[(a(),Q("svg",N,J)),z])]),G,X,$,t("p",null,[t("mjx-container",F,[(a(),Q("svg",q,K)),U]),e(" 값은 여러 개 존재할 수 있으며, 대각행렬로 lambda 값들을 표현할 수 있다. 대각행렬로 나타낸 람다값은 대문자 람다 "),t("mjx-container",Y,[(a(),Q("svg",t2,a2)),T2]),e("로 나타낸다.")]),t("p",null,[t("mjx-container",e2,[(a(),Q("svg",l2,o2)),n2]),e(" 로 나타낼 수 있는데, 고윳값을 갖는 모든 벡터는 Invertable 하다는 성질을 활용해 식을 정리하면")]),t("p",null,[t("mjx-container",d2,[(a(),Q("svg",m2,i2)),h2]),e(" 로 정리할 수 있다.")]),t("p",null,[e("이번에는 "),t("mjx-container",p2,[(a(),Q("svg",c2,H2)),w2]),e("만 남도록 식을 정리해보자. 마찬가지로, V의 invertable한 성질을 활용하도록 한다.")]),t("p",null,[t("mjx-container",u2,[(a(),Q("svg",x2,L2)),f2])]),t("p",null,[t("mjx-container",y2,[(a(),Q("svg",k2,V2)),Z2])]),t("p",null,[t("mjx-container",v2,[(a(),Q("svg",b2,j2)),C2])]),A2,t("ol",null,[t("li",null,[t("p",null,[t("strong",null,[t("mjx-container",S2,[(a(),Q("svg",R2,I2)),B2]),e("의 고유값은 A의 고유값과 같다.")])])]),t("li",null,[t("p",null,[t("strong",null,[e("A가 orthogonal matrix이면, "),t("mjx-container",O2,[(a(),Q("svg",N2,J2)),z2]),e(" 이다.")])]),t("p",null,[t("mjx-container",G2,[(a(),Q("svg",X2,F2)),q2])]),t("p",null,[t("mjx-container",W2,[(a(),Q("svg",K2,Y2)),t1])]),t("p",null,[t("mjx-container",Q1,[(a(),Q("svg",a1,e1)),l1])]),s1,t("p",null,[t("mjx-container",o1,[(a(),Q("svg",n1,m1)),r1])]),t("p",null,[t("mjx-container",i1,[(a(),Q("svg",h1,c1)),g1])]),H1]),t("li",null,[t("p",null,[t("strong",null,[e("A가 Postivie Semi Definite (PSD) 이면 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n개의 고유값을 갖는 경우 대각화 가능하다. 고유값분해와 혼동하지 말아야 할 것은, 고유값분해는 꼭 서로 다른 고유값을 가질 필요는 없다는 것이다.("),t("mjx-container",d3,[(a(),Q("svg",m3,i3)),h3]),e("의 대각성분으로 0이 대다수 나타나는 경우가 있다.)")])]),p3,c3,t("ol",null,[t("li",null,[g3,H3,t("p",null,[t("mjx-container",w3,[(a(),Q("svg",u3,_3)),L3])]),t("p",null,[e("여기서 "),t("mjx-container",f3,[(a(),Q("svg",y3,M3)),V3]),e("는 "),t("mjx-container",Z3,[(a(),Q("svg",v3,D3)),j3]),e("로, 나열된 수식에서 소거된다. 따라서, 이를 정리하면")]),t("p",null,[t("mjx-container",C3,[(a(),Q("svg",A3,R3)),P3])]),t("p",null,[e("을 얻을 수 있는데, "),t("mjx-container",I3,[(a(),Q("svg",B3,N3)),E3]),e("는 대각행렬의 제곱이므로 복잡한 연산 없이 대각성분들을 k승 해주기만 하면 된다.")])]),t("li",null,[J3,t("p",null,[t("mjx-container",z3,[(a(),Q("svg",G3,$3)),F3])]),t("p",null,[t("mjx-container",q3,[(a(),Q("svg",W3,U3)),Y3])])]),t("li",null,[t6,Q6,t("p",null,[t("mjx-container",a6,[(a(),Q("svg",T6,l6)),s6])]),t("p",null,[t("mjx-container",o6,[(a(),Q("svg",n6,m6)),r6])]),i6,t("p",null,[e("따라서, 결국에는 "),t("mjx-container",h6,[(a(),Q("svg",p6,g6)),H6]),e(" 만 남는다.")]),w6,t("p",null,[t("mjx-container",u6,[(a(),Q("svg",x6,L6)),f6])]),y6]),t("li",null,[k6,M6,t("p",null,[t("mjx-container",V6,[(a(),Q("svg",Z6,b6)),D6]),e("이고, 대각합의 성질 상, "),t("mjx-container",j6,[(a(),Q("svg",C6,S6)),R6]),e(" 로도 정리할 수 있다.")]),t("p",null,[t("mjx-container",P6,[(a(),Q("svg",I6,O6)),N6]),e(" 이므로, 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")],-1);function w7(u7,x7,_7,L7,f7,y7){return a(),Q("div",null,[o,t("p",null,[t("mjx-container",n,[(a(),Q("svg",d,r)),i])]),t("p",null,[e("여기서 "),t("mjx-container",h,[(a(),Q("svg",p,g)),H]),e(" 는 벡터에 곱해지는 스칼라를 의미한다. 이 스칼라 값인 람다를 "),w,e("이라고 칭한다. 즉, 고유값이란 고유벡터에 곱해지는 상수값을 의미한다.")]),u,t("p",null,[t("mjx-container",x,[(a(),Q("svg",_,f)),y])]),t("p",null,[t("mjx-container",k,[(a(),Q("svg",M,Z)),v])]),t("p",null,[t("mjx-container",b,[(a(),Q("svg",D,C)),A])]),t("p",null,[t("mjx-container",S,[(a(),Q("svg",R,I)),B])]),t("p",null,[t("mjx-container",O,[(a(),Q("svg",N,J)),z])]),G,X,$,t("p",null,[t("mjx-container",F,[(a(),Q("svg",q,K)),U]),e(" 값은 여러 개 존재할 수 있으며, 대각행렬로 lambda 값들을 표현할 수 있다. 대각행렬로 나타낸 람다값은 대문자 람다 "),t("mjx-container",Y,[(a(),Q("svg",t2,a2)),T2]),e("로 나타낸다.")]),t("p",null,[t("mjx-container",e2,[(a(),Q("svg",l2,o2)),n2]),e(" 로 나타낼 수 있는데, 고윳값을 갖는 모든 벡터는 Invertable 하다는 성질을 활용해 식을 정리하면")]),t("p",null,[t("mjx-container",d2,[(a(),Q("svg",m2,i2)),h2]),e(" 로 정리할 수 있다.")]),t("p",null,[e("이번에는 "),t("mjx-container",p2,[(a(),Q("svg",c2,H2)),w2]),e("만 남도록 식을 정리해보자. 마찬가지로, V의 invertable한 성질을 활용하도록 한다.")]),t("p",null,[t("mjx-container",u2,[(a(),Q("svg",x2,L2)),f2])]),t("p",null,[t("mjx-container",y2,[(a(),Q("svg",k2,V2)),Z2])]),t("p",null,[t("mjx-container",v2,[(a(),Q("svg",b2,j2)),C2])]),A2,t("ol",null,[t("li",null,[t("p",null,[t("strong",null,[t("mjx-container",S2,[(a(),Q("svg",R2,I2)),B2]),e("의 고유값은 A의 고유값과 같다.")])])]),t("li",null,[t("p",null,[t("strong",null,[e("A가 orthogonal matrix이면, "),t("mjx-container",O2,[(a(),Q("svg",N2,J2)),z2]),e(" 이다.")])]),t("p",null,[t("mjx-container",G2,[(a(),Q("svg",X2,F2)),q2])]),t("p",null,[t("mjx-container",W2,[(a(),Q("svg",K2,Y2)),t1])]),t("p",null,[t("mjx-container",Q1,[(a(),Q("svg",a1,e1)),l1])]),s1,t("p",null,[t("mjx-container",o1,[(a(),Q("svg",n1,m1)),r1])]),t("p",null,[t("mjx-container",i1,[(a(),Q("svg",h1,c1)),g1])]),H1]),t("li",null,[t("p",null,[t("strong",null,[e("A가 Postivie Semi Definite (PSD) 이면 "),t("mjx-container",w1,[(a(),Q("svg",u1,_1)),L1]),e("는 무조건 0보다 크거나 같다.")])])])]),f1,t("ol",y1,[t("li",null,[t("p",null,[t("strong",null,[e("Diagonal Matrix "),t("mjx-container",k1,[(a(),Q("svg",M1,Z1)),v1]),e("의 Non-Zero 고유값의 개수는 rank와 동일하다.")])]),b1,D1,j1,C1,t("p",null,[e("각 차원에 곱해진 "),t("mjx-container",A1,[(a(),Q("svg",S1,P1)),I1]),e("를 찾는 과정이다.")]),t("p",null,[e("여기서 "),t("mjx-container",B1,[(a(),Q("svg",O1,E1)),J1]),e("는 각 차원별로 곱해지는 값의 모음이므로, 대각행렬성분의 개수가 곧 고유벡터의 랭크와 같다.")]),z1]),t("li",null,[t("p",null,[t("strong",null,[e("Symmetric Matrix는 무조건 Diagonalizable 하며 (역 성립 X), 따라서 "),t("mjx-container",G1,[(a(),Q("svg",X1,F1)),q1]),e(" 된다.")])]),t("p",null,[e("대칭행렬이란 "),t("mjx-container",W1,[(a(),Q("svg",K1,Y1)),t3]),e("인 행렬이다. 대칭행렬은 무조건 대각화가 가능하다는 성질을 갖는다.")])])]),Q3,t("p",null,[e("대각화란 어떠한 행렬을 고유벡터로 이뤄진 가역행렬과 가역행렬에 곱해진 고유값들에 대한 대각행렬의 곱으로 나타내는 것을 의미한다. 즉, "),t("mjx-container",a3,[(a(),Q("svg",T3,l3)),s3]),e("의 꼴로 나타내는 것을 의미한다.")]),o3,t("ol",null,[n3,t("li",null,[e("n x n 행렬 A가 서로 다른 n개의 고유값을 갖는 경우 대각화 가능하다. 고유값분해와 혼동하지 말아야 할 것은, 고유값분해는 꼭 서로 다른 고유값을 가질 필요는 없다는 것이다.("),t("mjx-container",d3,[(a(),Q("svg",m3,i3)),h3]),e("의 대각성분으로 0이 대다수 나타나는 경우가 있다.)")])]),p3,c3,t("ol",null,[t("li",null,[g3,H3,t("p",null,[t("mjx-container",w3,[(a(),Q("svg",u3,_3)),L3])]),t("p",null,[e("여기서 "),t("mjx-container",f3,[(a(),Q("svg",y3,M3)),V3]),e("는 "),t("mjx-container",Z3,[(a(),Q("svg",v3,D3)),j3]),e("로, 나열된 수식에서 소거된다. 따라서, 이를 정리하면")]),t("p",null,[t("mjx-container",C3,[(a(),Q("svg",A3,R3)),P3])]),t("p",null,[e("을 얻을 수 있는데, "),t("mjx-container",I3,[(a(),Q("svg",B3,N3)),E3]),e("는 대각행렬의 제곱이므로 복잡한 연산 없이 대각성분들을 k승 해주기만 하면 된다.")])]),t("li",null,[J3,t("p",null,[t("mjx-container",z3,[(a(),Q("svg",G3,$3)),F3])]),t("p",null,[t("mjx-container",q3,[(a(),Q("svg",W3,U3)),Y3])])]),t("li",null,[t6,Q6,t("p",null,[t("mjx-container",a6,[(a(),Q("svg",T6,l6)),s6])]),t("p",null,[t("mjx-container",o6,[(a(),Q("svg",n6,m6)),r6])]),i6,t("p",null,[e("따라서, 결국에는 "),t("mjx-container",h6,[(a(),Q("svg",p6,g6)),H6]),e(" 만 남는다.")]),w6,t("p",null,[t("mjx-container",u6,[(a(),Q("svg",x6,L6)),f6])]),y6]),t("li",null,[k6,M6,t("p",null,[t("mjx-container",V6,[(a(),Q("svg",Z6,b6)),D6]),e("이고, 대각합의 성질 상, "),t("mjx-container",j6,[(a(),Q("svg",C6,S6)),R6]),e(" 로도 정리할 수 있다.")]),t("p",null,[t("mjx-container",P6,[(a(),Q("svg",I6,O6)),N6]),e(" 이므로, 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1. 고유값과 고유벡터 ​

고유벡터와 고유값 ​

고유하다는 것은 상황이 변화해도 그 특성을 잃지 않는 것을 의미한다. 그럼, 벡터가 고유하다는 것은 무엇일까? 벡터가 어떠한 상황에서도 그 특성, 즉 방향성을 잃지 않는 것을 의미한다. 즉, 고유벡터란 선형변환 이후에도 변환 결과가 자신의 상수배를 한 결과일 때의 벡터를 의미한다. 선형 변환이란 쉽게 말해 어떤 행렬을 벡터에 곱하는 것이다.

여러가지 선형변환들

선형변환이란?

벡터에 어떠한 행렬을 곱하는 것을 '벡터에 행렬을 통과시킨다'라고 표현한다. 아무튼, 벡터에 어떠한 행렬을 곱하게 되면 벡터의 크기와 방향이 변한다.

하지만, 아무리 벡터의 크기나 방향이 변해 봤자, 조금 더 벡터의 크기가 커진다거나, 벡터의 방향이 치우치는 정도에 그치게 된다. 벡터가 곡선의 형상을 띄거나 하지는 않는다는 것이다. 이를 '선형적으로 변화했다'라고 말한다.

이처럼, 벡터가 어떠한 행렬을 통과하여 선형적인 변화를 일으키게 하는 것을 선형변환 (linear transformation) 이라고 한다.

선형변환에 속하는 다양한 변환들이 있으며, 모든 변환들은 행렬을 곱하여 이뤄진다. 어찌보면, 벡터에 곱해지는 행렬이 곧 함수와도 같은 역할을 한다고 볼 수 있다. 이런 변환들은 주로 컴퓨터 그래픽에 많이 활용된다. 하지만 아핀변환이라는 것은 LLM에서도 자주 언급되긴 한다.

▶️scailing (비례변환) 비례변환은 벡터의 방향과 크기가 변화하는 변환을 의미한다.

▶️Rotation (회전변환) 회전변환은 좌표평면이 원점을 중심으로 회전하는 것을 의미한다. 회전변환된 벡터는 원래의 벡터와 선형독립이며, 회전변환시 고유값과 고유벡터는 존재하지 않는다.

▶️Shearing (전단변환) 전단변환은 특정 차원값에만 변화를 주는 변환을 의미한다. 기하학적으로 이해하면, y축을 고정하고 x축 방향으로만 변화를 가하는 것을 의미한다.

앞서 살펴본 변환들은 모두 원점이 변화하지 않는 변환이다. 원점을 이동시키는 변환도 있다. 바로 이동변환 이라는 것인데, 대표적으로 아핀변환(Affine) 이 있다.

아핀변환 (Affine Transformation) TBD

식으로 나타내면 다음과 같이 표현할 수 있다.

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")],-1);function w7(u7,x7,_7,L7,f7,y7){return a(),Q("div",null,[o,t("p",null,[t("mjx-container",n,[(a(),Q("svg",d,r)),i])]),t("p",null,[e("여기서 "),t("mjx-container",h,[(a(),Q("svg",p,g)),H]),e(" 는 벡터에 곱해지는 스칼라를 의미한다. 이 스칼라 값인 람다를 "),w,e("이라고 칭한다. 즉, 고유값이란 고유벡터에 곱해지는 상수값을 의미한다.")]),u,t("p",null,[t("mjx-container",x,[(a(),Q("svg",_,f)),y])]),t("p",null,[t("mjx-container",k,[(a(),Q("svg",M,Z)),v])]),t("p",null,[t("mjx-container",b,[(a(),Q("svg",D,C)),A])]),t("p",null,[t("mjx-container",S,[(a(),Q("svg",R,I)),B])]),t("p",null,[t("mjx-container",O,[(a(),Q("svg",N,J)),z])]),G,X,$,t("p",null,[t("mjx-container",F,[(a(),Q("svg",q,K)),U]),e(" 값은 여러 개 존재할 수 있으며, 대각행렬로 lambda 값들을 표현할 수 있다. 대각행렬로 나타낸 람다값은 대문자 람다 "),t("mjx-container",Y,[(a(),Q("svg",t2,a2)),T2]),e("로 나타낸다.")]),t("p",null,[t("mjx-container",e2,[(a(),Q("svg",l2,o2)),n2]),e(" 로 나타낼 수 있는데, 고윳값을 갖는 모든 벡터는 Invertable 하다는 성질을 활용해 식을 정리하면")]),t("p",null,[t("mjx-container",d2,[(a(),Q("svg",m2,i2)),h2]),e(" 로 정리할 수 있다.")]),t("p",null,[e("이번에는 "),t("mjx-container",p2,[(a(),Q("svg",c2,H2)),w2]),e("만 남도록 식을 정리해보자. 마찬가지로, V의 invertable한 성질을 활용하도록 한다.")]),t("p",null,[t("mjx-container",u2,[(a(),Q("svg",x2,L2)),f2])]),t("p",null,[t("mjx-container",y2,[(a(),Q("svg",k2,V2)),Z2])]),t("p",null,[t("mjx-container",v2,[(a(),Q("svg",b2,j2)),C2])]),A2,t("ol",null,[t("li",null,[t("p",null,[t("strong",null,[t("mjx-container",S2,[(a(),Q("svg",R2,I2)),B2]),e("의 고유값은 A의 고유값과 같다.")])])]),t("li",null,[t("p",null,[t("strong",null,[e("A가 orthogonal matrix이면, "),t("mjx-container",O2,[(a(),Q("svg",N2,J2)),z2]),e(" 이다.")])]),t("p",null,[t("mjx-container",G2,[(a(),Q("svg",X2,F2)),q2])]),t("p",null,[t("mjx-container",W2,[(a(),Q("svg",K2,Y2)),t1])]),t("p",null,[t("mjx-container",Q1,[(a(),Q("svg",a1,e1)),l1])]),s1,t("p",null,[t("mjx-container",o1,[(a(),Q("svg",n1,m1)),r1])]),t("p",null,[t("mjx-container",i1,[(a(),Q("svg",h1,c1)),g1])]),H1]),t("li",null,[t("p",null,[t("strong",null,[e("A가 Postivie Semi Definite (PSD) 이면 "),t("mjx-container",w1,[(a(),Q("svg",u1,_1)),L1]),e("는 무조건 0보다 크거나 같다.")])])])]),f1,t("ol",y1,[t("li",null,[t("p",null,[t("strong",null,[e("Diagonal Matrix "),t("mjx-container",k1,[(a(),Q("svg",M1,Z1)),v1]),e("의 Non-Zero 고유값의 개수는 rank와 동일하다.")])]),b1,D1,j1,C1,t("p",null,[e("각 차원에 곱해진 "),t("mjx-container",A1,[(a(),Q("svg",S1,P1)),I1]),e("를 찾는 과정이다.")]),t("p",null,[e("여기서 "),t("mjx-container",B1,[(a(),Q("svg",O1,E1)),J1]),e("는 각 차원별로 곱해지는 값의 모음이므로, 대각행렬성분의 개수가 곧 고유벡터의 랭크와 같다.")]),z1]),t("li",null,[t("p",null,[t("strong",null,[e("Symmetric Matrix는 무조건 Diagonalizable 하며 (역 성립 X), 따라서 "),t("mjx-container",G1,[(a(),Q("svg",X1,F1)),q1]),e(" 된다.")])]),t("p",null,[e("대칭행렬이란 "),t("mjx-container",W1,[(a(),Q("svg",K1,Y1)),t3]),e("인 행렬이다. 대칭행렬은 무조건 대각화가 가능하다는 성질을 갖는다.")])])]),Q3,t("p",null,[e("대각화란 어떠한 행렬을 고유벡터로 이뤄진 가역행렬과 가역행렬에 곱해진 고유값들에 대한 대각행렬의 곱으로 나타내는 것을 의미한다. 즉, "),t("mjx-container",a3,[(a(),Q("svg",T3,l3)),s3]),e("의 꼴로 나타내는 것을 의미한다.")]),o3,t("ol",null,[n3,t("li",null,[e("n x n 행렬 A가 서로 다른 n개의 고유값을 갖는 경우 대각화 가능하다. 고유값분해와 혼동하지 말아야 할 것은, 고유값분해는 꼭 서로 다른 고유값을 가질 필요는 없다는 것이다.("),t("mjx-container",d3,[(a(),Q("svg",m3,i3)),h3]),e("의 대각성분으로 0이 대다수 나타나는 경우가 있다.)")])]),p3,c3,t("ol",null,[t("li",null,[g3,H3,t("p",null,[t("mjx-container",w3,[(a(),Q("svg",u3,_3)),L3])]),t("p",null,[e("여기서 "),t("mjx-container",f3,[(a(),Q("svg",y3,M3)),V3]),e("는 "),t("mjx-container",Z3,[(a(),Q("svg",v3,D3)),j3]),e("로, 나열된 수식에서 소거된다. 따라서, 이를 정리하면")]),t("p",null,[t("mjx-container",C3,[(a(),Q("svg",A3,R3)),P3])]),t("p",null,[e("을 얻을 수 있는데, "),t("mjx-container",I3,[(a(),Q("svg",B3,N3)),E3]),e("는 대각행렬의 제곱이므로 복잡한 연산 없이 대각성분들을 k승 해주기만 하면 된다.")])]),t("li",null,[J3,t("p",null,[t("mjx-container",z3,[(a(),Q("svg",G3,$3)),F3])]),t("p",null,[t("mjx-container",q3,[(a(),Q("svg",W3,U3)),Y3])])]),t("li",null,[t6,Q6,t("p",null,[t("mjx-container",a6,[(a(),Q("svg",T6,l6)),s6])]),t("p",null,[t("mjx-container",o6,[(a(),Q("svg",n6,m6)),r6])]),i6,t("p",null,[e("따라서, 결국에는 "),t("mjx-container",h6,[(a(),Q("svg",p6,g6)),H6]),e(" 만 남는다.")]),w6,t("p",null,[t("mjx-container",u6,[(a(),Q("svg",x6,L6)),f6])]),y6]),t("li",null,[k6,M6,t("p",null,[t("mjx-container",V6,[(a(),Q("svg",Z6,b6)),D6]),e("이고, 대각합의 성질 상, "),t("mjx-container",j6,[(a(),Q("svg",C6,S6)),R6]),e(" 로도 정리할 수 있다.")]),t("p",null,[t("mjx-container",P6,[(a(),Q("svg",I6,O6)),N6]),e(" 이므로, 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Definte란?")],-1),y1={start:"4"},k1={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},M1={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"1.57ex",height:"1.62ex",role:"img",focusable:"false",viewBox:"0 -716 694 716","aria-hidden":"true"},V1=t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"39B",d:"M320 708Q326 716 340 716H348H355Q367 716 372 708Q374 706 423 547T523 226T575 62Q581 52 591 50T634 46H661V0H653Q644 3 532 3Q411 3 390 0H379V46H392Q464 46 464 65Q463 70 390 305T316 539L246 316Q177 95 177 84Q177 72 198 59T248 46H253V0H245Q230 3 130 3Q47 3 38 0H32V46H45Q112 51 127 91Q128 92 224 399T320 708Z",style:{"stroke-width":"3"}})])])],-1),Z1=[V1],v1=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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")],-1);function w7(u7,x7,_7,L7,f7,y7){return a(),Q("div",null,[o,t("p",null,[t("mjx-container",n,[(a(),Q("svg",d,r)),i])]),t("p",null,[e("여기서 "),t("mjx-container",h,[(a(),Q("svg",p,g)),H]),e(" 는 벡터에 곱해지는 스칼라를 의미한다. 이 스칼라 값인 람다를 "),w,e("이라고 칭한다. 즉, 고유값이란 고유벡터에 곱해지는 상수값을 의미한다.")]),u,t("p",null,[t("mjx-container",x,[(a(),Q("svg",_,f)),y])]),t("p",null,[t("mjx-container",k,[(a(),Q("svg",M,Z)),v])]),t("p",null,[t("mjx-container",b,[(a(),Q("svg",D,C)),A])]),t("p",null,[t("mjx-container",S,[(a(),Q("svg",R,I)),B])]),t("p",null,[t("mjx-container",O,[(a(),Q("svg",N,J)),z])]),G,X,$,t("p",null,[t("mjx-container",F,[(a(),Q("svg",q,K)),U]),e(" 값은 여러 개 존재할 수 있으며, 대각행렬로 lambda 값들을 표현할 수 있다. 대각행렬로 나타낸 람다값은 대문자 람다 "),t("mjx-container",Y,[(a(),Q("svg",t2,a2)),T2]),e("로 나타낸다.")]),t("p",null,[t("mjx-container",e2,[(a(),Q("svg",l2,o2)),n2]),e(" 로 나타낼 수 있는데, 고윳값을 갖는 모든 벡터는 Invertable 하다는 성질을 활용해 식을 정리하면")]),t("p",null,[t("mjx-container",d2,[(a(),Q("svg",m2,i2)),h2]),e(" 로 정리할 수 있다.")]),t("p",null,[e("이번에는 "),t("mjx-container",p2,[(a(),Q("svg",c2,H2)),w2]),e("만 남도록 식을 정리해보자. 마찬가지로, V의 invertable한 성질을 활용하도록 한다.")]),t("p",null,[t("mjx-container",u2,[(a(),Q("svg",x2,L2)),f2])]),t("p",null,[t("mjx-container",y2,[(a(),Q("svg",k2,V2)),Z2])]),t("p",null,[t("mjx-container",v2,[(a(),Q("svg",b2,j2)),C2])]),A2,t("ol",null,[t("li",null,[t("p",null,[t("strong",null,[t("mjx-container",S2,[(a(),Q("svg",R2,I2)),B2]),e("의 고유값은 A의 고유값과 같다.")])])]),t("li",null,[t("p",null,[t("strong",null,[e("A가 orthogonal matrix이면, "),t("mjx-container",O2,[(a(),Q("svg",N2,J2)),z2]),e(" 이다.")])]),t("p",null,[t("mjx-container",G2,[(a(),Q("svg",X2,F2)),q2])]),t("p",null,[t("mjx-container",W2,[(a(),Q("svg",K2,Y2)),t1])]),t("p",null,[t("mjx-container",Q1,[(a(),Q("svg",a1,e1)),l1])]),s1,t("p",null,[t("mjx-container",o1,[(a(),Q("svg",n1,m1)),r1])]),t("p",null,[t("mjx-container",i1,[(a(),Q("svg",h1,c1)),g1])]),H1]),t("li",null,[t("p",null,[t("strong",null,[e("A가 Postivie Semi Definite (PSD) 이면 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n개의 고유값을 갖는 경우 대각화 가능하다. 고유값분해와 혼동하지 말아야 할 것은, 고유값분해는 꼭 서로 다른 고유값을 가질 필요는 없다는 것이다.("),t("mjx-container",d3,[(a(),Q("svg",m3,i3)),h3]),e("의 대각성분으로 0이 대다수 나타나는 경우가 있다.)")])]),p3,c3,t("ol",null,[t("li",null,[g3,H3,t("p",null,[t("mjx-container",w3,[(a(),Q("svg",u3,_3)),L3])]),t("p",null,[e("여기서 "),t("mjx-container",f3,[(a(),Q("svg",y3,M3)),V3]),e("는 "),t("mjx-container",Z3,[(a(),Q("svg",v3,D3)),j3]),e("로, 나열된 수식에서 소거된다. 따라서, 이를 정리하면")]),t("p",null,[t("mjx-container",C3,[(a(),Q("svg",A3,R3)),P3])]),t("p",null,[e("을 얻을 수 있는데, "),t("mjx-container",I3,[(a(),Q("svg",B3,N3)),E3]),e("는 대각행렬의 제곱이므로 복잡한 연산 없이 대각성분들을 k승 해주기만 하면 된다.")])]),t("li",null,[J3,t("p",null,[t("mjx-container",z3,[(a(),Q("svg",G3,$3)),F3])]),t("p",null,[t("mjx-container",q3,[(a(),Q("svg",W3,U3)),Y3])])]),t("li",null,[t6,Q6,t("p",null,[t("mjx-container",a6,[(a(),Q("svg",T6,l6)),s6])]),t("p",null,[t("mjx-container",o6,[(a(),Q("svg",n6,m6)),r6])]),i6,t("p",null,[e("따라서, 결국에는 "),t("mjx-container",h6,[(a(),Q("svg",p6,g6)),H6]),e(" 만 남는다.")]),w6,t("p",null,[t("mjx-container",u6,[(a(),Q("svg",x6,L6)),f6])]),y6]),t("li",null,[k6,M6,t("p",null,[t("mjx-container",V6,[(a(),Q("svg",Z6,b6)),D6]),e("이고, 대각합의 성질 상, "),t("mjx-container",j6,[(a(),Q("svg",C6,S6)),R6]),e(" 로도 정리할 수 있다.")]),t("p",null,[t("mjx-container",P6,[(a(),Q("svg",I6,O6)),N6]),e(" 이므로, 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0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"W")])],-1),w=t("blockquote",null,[t("p",null,"예를들어 키, 몸무게에 따른 남/여 카테고리 컬럼이 존재하는 경우, 남성과 여성을 구분하는 결정 직선을 구하고자 한다면 로지스틱 회귀를 적용해야 한다.")],-1),u=t("blockquote",null,[t("p",null,"최소제곱오차에 따른 선형 회귀 방식을 적용하는 경우는 키에 따른 몸무게를 예측하고자 하는 상황과 같이, 이미 존재하는 두 수치 변수간의 관계를 파악하고자 할 때 이다.")],-1),x=t("p",null,"위의 사례들에서 알 수 있다시피, 특정 입력 값에 대한 출력값을 예측하는 모델(function)을 구하고자 할 때는 선형회귀를, 레이블이 존재 할 때 입력값들을 구분하고자 하는 기준 function을 구하고자 할 때는 로지스틱 회귀를 적용한다.",-1),f=t("p",null,"그런데 이제, 선형회귀에서는 결정 공간(치역)이 연속적인 값들로 구성되어 있지만, 로지스틱에서는 결정 공간이 레이블 값(흔히 0, 1)로만 구성되어 있다. 그래서 로지스틱 회귀에서는 시그모이드 함수에 사상된 벡터를 넣어서 0~1사이의 값으로 바꿔버린다.",-1),y={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},L={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.186ex"},xmlns:"http://www.w3.org/2000/svg",width:"6.65ex",height:"1.756ex",role:"img",focusable:"false",viewBox:"0 -694 2939.4 776","aria-hidden":"true"},M=e('',1),k=[M],Z=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"w"),t("mi",null,"t")]),t("mi",null,"x")])],-1),K={class:"info custom-block"},Y=t("p",{class:"custom-block-title"},"💡시그모이드 함수와 우도",-1),t1=t("h3",{id:"시그모이드-함수",tabindex:"-1"},[a("시그모이드 함수 "),t("a",{class:"header-anchor",href:"#시그모이드-함수","aria-label":'Permalink to "시그모이드 함수"'},"​")],-1),T1=t("p",null,"시그모이드 함수 역시 입력 값을 특정 값으로 사상시키는 함수로, 전체 실수 값을 0~1 사이의 값으로 변환해주는 함수이다.",-1),Q1={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},a1={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.172ex"},xmlns:"http://www.w3.org/2000/svg",width:"36.619ex",height:"5.208ex",role:"img",focusable:"false",viewBox:"0 -1342 16185.7 2302","aria-hidden":"true"},e1=e('',1),l1=[e1],s1=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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454","aria-hidden":"true"},d1=t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D464",d:"M580 385Q580 406 599 424T641 443Q659 443 674 425T690 368Q690 339 671 253Q656 197 644 161T609 80T554 12T482 -11Q438 -11 404 5T355 48Q354 47 352 44Q311 -11 252 -11Q226 -11 202 -5T155 14T118 53T104 116Q104 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Q21 293 29 315T52 366T96 418T161 441Q204 441 227 416T250 358Q250 340 217 250T184 111Q184 65 205 46T258 26Q301 26 334 87L339 96V119Q339 122 339 128T340 136T341 143T342 152T345 165T348 182T354 206T362 238T373 281Q402 395 406 404Q419 431 449 431Q468 431 475 421T483 402Q483 389 454 274T422 142Q420 131 420 107V100Q420 85 423 71T442 42T487 26Q558 26 600 148Q609 171 620 213T632 273Q632 306 619 325T593 357T580 385Z",style:{"stroke-width":"3"}})])])],-1),n1=[d1],r1=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"w")])],-1),h1={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},i1={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"0.971ex",height:"1.595ex",role:"img",focusable:"false",viewBox:"0 -694 429 705","aria-hidden":"true"},p1=t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D44F",d:"M73 647Q73 657 77 670T89 683Q90 683 161 688T234 694Q246 694 246 685T212 542Q204 508 195 472T180 418L176 399Q176 396 182 402Q231 442 283 442Q345 442 383 396T422 280Q422 169 343 79T173 -11Q123 -11 82 27T40 150V159Q40 180 48 217T97 414Q147 611 147 623T109 637Q104 637 101 637H96Q86 637 83 637T76 640T73 647ZM336 325V331Q336 405 275 405Q258 405 240 397T207 376T181 352T163 330L157 322L136 236Q114 150 114 114Q114 66 138 42Q154 26 178 26Q211 26 245 58Q270 81 285 114T318 219Q336 291 336 325Z",style:{"stroke-width":"3"}})])])],-1),g1=[p1],c1=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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경우라도 지수값이 0이 되면서 해당 시그모이드 값이 무시되고 반대의 확률값만 곱해지게 된다.",-1),N1={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},z1={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"4.921ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 2175 1000","aria-hidden":"true"},G1=e("",1),J1=[G1],F1=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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🌱 선형대수-기초 ​

추가 공부자료 ​

인공지능을 위한 선형대수

선형대수학 | 수학 | Khan Academy

혁펜하임의 "보이는" 선형대수학 (Linear Algebra)

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🌱 선형대수-기초 ​

추가 공부자료 ​

인공지능을 위한 선형대수

선형대수학 | 수학 | Khan Academy

혁펜하임의 "보이는" 선형대수학 (Linear Algebra)

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곱"})],-1),W=t("p",null,"이러한 단위 행렬의 성질은 역행렬과 직교행렬의 특성을 설명할 때 다시 언급된다.",-1),U=t("h2",{id:"대각행렬-diagonal-matrix",tabindex:"-1"},[Q("대각행렬(diagonal matrix) "),t("a",{class:"header-anchor",href:"#대각행렬-diagonal-matrix","aria-label":'Permalink to "대각행렬(diagonal matrix)"'},"​")],-1),Y=t("p",null,"대각행렬은 대각 성분을 제외한 원소들이 모두 0이라는 점에서는 동일하나, 대각성분이 1이 아닌 값도 될 수 있다. 즉, 단위 행렬은 대각 행렬의 일종이라고도 이해할 수 있다.",-1),t1={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},Q1={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.025ex"},xmlns:"http://www.w3.org/2000/svg",width:"1.097ex",height:"1.027ex",role:"img",focusable:"false",viewBox:"0 -443 485 454","aria-hidden":"true"},a1=t("g",{stroke:"currentColor",fill:"currentColor","stroke-width":"0",transform:"scale(1,-1)"},[t("g",{"data-mml-node":"math"},[t("g",{"data-mml-node":"mi"},[t("path",{"data-c":"1D463",d:"M173 380Q173 405 154 405Q130 405 104 376T61 287Q60 286 59 284T58 281T56 279T53 278T49 278T41 278H27Q21 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833.9","aria-hidden":"true"},z1=e('',1),G1=[z1],q1=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"A"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"−"),t("mn",null,"1")])])])],-1),F1=t("p",null,"역행렬을 본격적으로 이해해보기 전에, 헹렬식에 대해 알아보자.",-1),K1=t("h3",{id:"행렬식-determinant",tabindex:"-1"},[Q("행렬식 (determinant) "),t("a",{class:"header-anchor",href:"#행렬식-determinant","aria-label":'Permalink to "행렬식 (determinant)"'},"​")],-1),$1=t("p",null,"행렬식이란, 정방행렬에 하나의 수를 대응시키는 함수를 의미한다. 행렬을 통해 연립일차방정식의 해(크래머 공식)를 구하기 위해 고안되었다고 한다. 그 외에도 고윳값을 계산할 떄도 등장하는 용어다. 역행렬에서 행렬식에 대해 다루는 이유는, 행렬식을 통해 역행렬의 존재성을 판별하기 때문이다.",-1),W1=t("p",null,[Q("이차정사각행렬에서 기하학적으로 행렬식을 이해하면, 평면상에서 행렬의 열벡터를 표현했을때, 각각의 종점을 연장하여 평행사변형을 만들었을 때의 넓이가 행렬식의 값과 같다. "),t("img",{src:n,alt:"정사각행렬의 열벡터로 나타낸 평행사변형"})],-1),U1=t("p",null,"삼차정사각행렬에서의 행렬식의 값도 마찬가지로, 3차원 공간상에 표현된 평행육면체의 부피와 같다. 2X2 정방행렬에서의 행렬식은 다음과 같다",-1),Y1={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},t2={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.149ex"},xmlns:"http://www.w3.org/2000/svg",width:"20.556ex",height:"5.43ex",role:"img",focusable:"false",viewBox:"0 -1450 9085.7 2400","aria-hidden":"true"},Q2=e('',1),a2=[Q2],T2=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"d"),t("mi",null,"e"),t("mi",null,"t"),t("mrow",{"data-mjx-texclass":"INNER"},[t("mo",{"data-mjx-texclass":"OPEN"},"["),t("mtable",{columnspacing:"1em",rowspacing:"4pt"},[t("mtr",null,[t("mtd",null,[t("mi",null,"a")]),t("mtd",null,[t("mi",null,"b")])]),t("mtr",null,[t("mtd",null,[t("mi",null,"c")]),t("mtd",null,[t("mi",null,"d")])])]),t("mo",{"data-mjx-texclass":"CLOSE"},"]")]),t("mo",null,"="),t("mi",null,"a"),t("mi",null,"d"),t("mo",null,"−"),t("mi",null,"b"),t("mi",null,"c")])],-1),e2=e('

2X2 정방행렬에서의 행렬식 연산은 간단하다. 대각선을 이루는 원소들끼리 곱하고, 이들을 더하기만 하면 된다.

3X3 정방행렬에서의 행렬식 연산은 좀 복잡하다. 두 가지 방법으로 구해볼 수 있다.

전개 방식

행렬에서 마지막 열을 제외한 나머지 열을 마지막 열의 다음에 붙여준다

이 상태에서 다음 연산을 진행해준다.

3X3 행렬식 1

그 다음엔 반대로 구한다.

3X3 행렬식 2

마지막으로 앞서 구한 두 값을 뺀다.

3X3 행렬식 3

여인수와 소행렬 ​

앞서 짧게 소개한 전개 방식은 사실 여인수전개를 통한 계산을 간단히 한 것이다.

소행렬을 통한 계산을 위해선 소행렬과 여인수전개의 개념을 먼저 숙지해야 한다.

소행렬 (minor determinant)

소행렬이란, 특정 열과 행을 제거하고 만든 부분행렬에 대한 행렬식을 의미한다.

행렬에 제외되는 행, 열을 아래첨자로 표기하면 된다. 또는 소행렬에 절댓값 기호를 취해서 나타내는 방법도 있다.

위 식은 i행, j열의 원소들을 제하고 남은 부분에 대한 행렬식을 의미한다. 그림으로 나타내면 다음과 같다.

소행렬

여인수(cofactor)와 여인수전개

여인수전개는 여인수로 행렬식을 구하는 방법을 의미한다. 라플라스 전개라고 부르기도 한다.

여인수는 소행렬에 (-1)^(i+j)를 곱한 값을 의미한다. 식으로 나타내면 다음과 같다.

',14),l2={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},s2={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.666ex"},xmlns:"http://www.w3.org/2000/svg",width:"47.239ex",height:"2.8ex",role:"img",focusable:"false",viewBox:"0 -943.3 20879.4 1237.6","aria-hidden":"true"},o2=e('',1),n2=[o2],d2=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"C"),t("mo",null,"="),t("mi",null,"C"),t("mi",null,"o"),t("mi",null,"f"),t("mi",null,"a"),t("mi",null,"t"),t("mi",null,"o"),t("mi",null,"r"),t("mo",null,","),t("msub",null,[t("mi",null,"A"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mi",null,"j")])]),t("mo",null,"="),t("mi",null,"S"),t("mi",null,"u"),t("mi",null,"b"),t("mi",null,"m"),t("mi",null,"a"),t("mi",null,"t"),t("mi",null,"r"),t("mi",null,"i"),t("mi",null,"x"),t("mspace",{linebreak:"newline"}),t("mspace",{linebreak:"newline"}),t("mi",null,"C"),t("mo",null,"="),t("mo",{stretchy:"false"},"("),t("mo",null,"−"),t("mn",null,"1"),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",{stretchy:"false"},"("),t("mi",null,"i"),t("mo",null,"+"),t("mi",null,"j"),t("mo",{stretchy:"false"},")")])]),t("msub",null,[t("mi",null,"A"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mi",null,"j")])])])],-1),m2=t("p",null,"여인수 전개에서는 어떠한 행, 열에 대해 여인수 전개를 진행해도 동일한 값이 도출된다. 따라서, 한 행이나 열이 모두 동일한 값이거나 0이 많이 포함된 경우, 이 행/열에 대해 여인수 전개를 진행하면 매우 효율적으로 계산을 진행할 수 있다.",-1),r2=t("p",null,"여인수전개는 4x4, 5x5 등 모든 차원의 정방행렬에 대해 적용될 수 있다.",-1),i2=t("p",null,"다음은 3X3 행렬에서 여인수전개로 행렬식 값을 구한 과정을 나타낸 그림이다.",-1),h2=t("p",null,[t("img",{src:h,alt:"여인수 전개"})],-1),p2=t("p",null,"위 그림대로 식들을 전개하면 다음의 식을 얻을 수 있다.",-1),c2={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},g2={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"63.037ex",height:"2.7ex",role:"img",focusable:"false",viewBox:"0 -943.3 27862.3 1193.3","aria-hidden":"true"},H2=e('',1),u2=[H2],w2=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msub",null,[t("mi",null,"a"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"21")])]),t("mo",null,"∗"),t("mo",{stretchy:"false"},"("),t("mo",null,"−"),t("mn",null,"1"),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",{stretchy:"false"},"("),t("mn",null,"2"),t("mo",null,"+"),t("mn",null,"1"),t("mo",{stretchy:"false"},")")])]),t("mo",null,"⋅"),t("msub",null,[t("mi",null,"A"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"21")])]),t("mo",null,"+"),t("msub",null,[t("mi",null,"a"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"22")])]),t("mo",null,"∗"),t("mo",{stretchy:"false"},"("),t("mo",null,"−"),t("mn",null,"1"),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",{stretchy:"false"},"("),t("mn",null,"2"),t("mo",null,"+"),t("mn",null,"2"),t("mo",{stretchy:"false"},")")])]),t("mo",null,"⋅"),t("msub",null,[t("mi",null,"A"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"22")])]),t("mo",null,"+"),t("msub",null,[t("mi",null,"a"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"23")])]),t("mo",null,"∗"),t("mo",{stretchy:"false"},"("),t("mo",null,"−"),t("mn",null,"1"),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",{stretchy:"false"},"("),t("mn",null,"2"),t("mo",null,"+"),t("mn",null,"3"),t("mo",{stretchy:"false"},")")])]),t("mo",null,"⋅"),t("msub",null,[t("mi",null,"A"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"23")])])])],-1),x2=t("p",null,"실제 값을 대입하면 다음의 결과를 얻을 수 있다.",-1),L2={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},_2={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"78.061ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 34503.1 1000","aria-hidden":"true"},y2=e('',1),f2=[y2],M2=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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코드상으로 구현하면, 5X5 에서 4X4, 3X3 ... 으로 점차 줄어들어 쉽게 계산할 수 있는 2X2의 소행렬로 나눠 계산한 후, 다른 계산 결과와 합쳐 나가는 방식을 취할 것이다. 이처럼 자기 자신을 더 작게 나누어 계산 가능한 사이즈로 나눠 계산한 후 원래의 상태로 거슬러 올라가며 합쳐나가는 알고리즘은 재귀적 프로그래밍으로 구현했을 때 효과적이다. 하지만 대체로 텐서 연산 관련 라이브러리에 잘 구현되어 있다.

파이썬 상에서 행렬식은 구하고자 한다면, 복잡한 구현 없이 다음의 코드로 쉽게 구할 수 있다.

X = np.array([[1,2,4],[2,-1,3],[0,5,1]])
+np.linalg.det(X)

행렬식과 역행렬 ​

행렬식을 알아본 이유는, 행렬식을 통해 정방행렬에 역행렬이 존재하는지를 확인할 수 있기 때문이다.

`,5),V2=t("strong",null,"역행렬이 존재하기 위해선, 행렬식의 값이 0이 아니어야 한다.",-1),Z2={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},b2={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"10.653ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 4708.6 1000","aria-hidden":"true"},v2=e('',1),D2=[v2],j2=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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직교행렬과 직교행렬의 곱은 직교행렬이다. 이는 직교행렬이 각도와 길이, 내적을 보존하는 행렬이기 때문이다.

Ref ​

[선형대수학] Determinant / 행렬식

행렬의 성분, 두 행렬이 서로 같을 조건

행렬식 이 영상만 보면 기본 끝! | 행렬식의 성질 | 기본행연산과 행렬식

orthogonal matrix(직교행렬) 과 matrix of orthonormal columns

',6);function k3(V3,Z3,b3,v3,D3,j3){return T(),a("div",null,[c,g,t("p",null,[Q("대칭 행렬은 자신의 전치 행렬이 원래의 자기 자신과 같은 행렬이다. 즉, "),t("mjx-container",H,[(T(),a("svg",u,x)),L]),Q(" 인 행렬을 의미한다.")]),t("p",null,[Q("대칭행렬을 이루기 위해선 원소의 row index와 column index가 반대로 바뀌어도 동일한 원소를 가져야 한다. "),t("mjx-container",_,[(T(),a("svg",y,M)),k]),Q(" 라는 것이다.")]),V,t("p",null,[Q("여기서, "),t("mjx-container",Z,[(T(),a("svg",b,D)),j]),Q(" 인 원소들을 "),C,Q(" 이라고 한다. 대칭행렬은 대각성분을 기준으로 대칭을 이룬다.")]),A,S,R,O,t("p",null,[Q("단위행렬은 "),t("mjx-container",P,[(T(),a("svg",E,B)),X]),Q("으로 나타낸다. n은 행(열)의 개수를 의미한다. 벡터에 단위 연산을 곱하는 연산을 수식으로 나타내면 다음과 같다")]),t("mjx-container",N,[(T(),a("svg",J,G)),q]),F,K,$,W,U,Y,t("p",null,[Q("대각성분벡터(주대각선)로 대각행렬을 표기하는 방법은 다음과 같다. 이때 "),t("mjx-container",t1,[(T(),a("svg",Q1,T1)),e1]),Q("는 주대각선을 이루는 대각성분벡터이다.")]),t("mjx-container",l1,[(T(),a("svg",s1,n1)),d1]),t("p",null,[Q("대각행렬의 대각성분을 이루는 벡터("),t("mjx-container",m1,[(T(),a("svg",r1,h1)),p1]),Q(")와 벡터의 곱셈연산은 아마다르곱("),t("mjx-container",c1,[(T(),a("svg",g1,u1)),w1]),Q(")과 같다.")]),t("mjx-container",x1,[(T(),a("svg",L1,y1)),f1]),M1,t("p",null,[Q("역행렬은 같은 꼴의 정방행렬 "),t("mjx-container",k1,[(T(),a("svg",V1,b1)),v1]),Q("와 단위행렬 "),t("mjx-container",D1,[(T(),a("svg",j1,A1)),S1]),Q("에 대해 "),t("mjx-container",R1,[(T(),a("svg",O1,E1)),I1]),Q("를 만족시키는 행렬을 의미한다. 일반적으로 행렬곱은 교환법칙이 성립하지 않아 곱셉 순서를 바꾸면 그 결과가 달라지나, 역행렬의 경우 곱셈의 순서를 바꾸어도 그 결과가 단위행렬로 항상 동일하다.")]),B1,X1,t("p",null,[Q("역행렬에 대한 표기는 "),t("mjx-container",N1,[(T(),a("svg",J1,G1)),q1]),Q("로 한다. 분수의 표기와 동일하다. 단위행렬과 행렬의 곱이 스칼라 1을 행렬에 곱한 결과와 비슷하다는 점에서 분수가 연상되기도 한다.")]),F1,K1,$1,W1,U1,t("mjx-container",Y1,[(T(),a("svg",t2,a2)),T2]),e2,t("mjx-container",l2,[(T(),a("svg",s2,n2)),d2]),m2,r2,i2,h2,p2,t("mjx-container",c2,[(T(),a("svg",g2,u2)),w2]),x2,t("mjx-container",L2,[(T(),a("svg",_2,f2)),M2]),k2,t("p",null,[V2,Q(" 즉, "),t("mjx-container",Z2,[(T(),a("svg",b2,D2)),j2]),Q("이 역행렬의 성립조건이다. 앞서 언급했듯이, 행렬식의 값은 벡터의 종점을 벡터의 크기만큼 연장시킨 평행사변형의 넓이이다. 행렬값이 0이라는 것은 다시 말하면, 두 벡터가 한 직선 상에 놓인 것이나 마찬가지이다.")]),C2,t("mjx-container",A2,[(T(),a("svg",S2,O2)),P2]),E2,I2,B2,t("p",null,[Q("또한, 직교행렬의 전치행렬과 직교 행렬의 곱은 단위행렬이다. 즉, "),t("mjx-container",X2,[(T(),a("svg",N2,z2)),G2]),Q(" 이다. 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통해 역행렬의 존재성을 판별하기 때문이다.",-1),W1=t("p",null,[Q("이차정사각행렬에서 기하학적으로 행렬식을 이해하면, 평면상에서 행렬의 열벡터를 표현했을때, 각각의 종점을 연장하여 평행사변형을 만들었을 때의 넓이가 행렬식의 값과 같다. 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1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("mi",null,"A"),t("mi",null,"X"),t("mo",null,"="),t("mi",null,"X"),t("mi",null,"A"),t("mo",null,"="),t("mi",null,"I")])],-1),B1=t("p",null,"역행렬은 정방행렬이어야 하며, 어떠한 정방행렬에 있어 역행렬은 오직 하나만 존재한다.",-1),X1=t("p",null,[Q("모든 정방행렬이 역행렬을 가지는 것은 아니며, 이러한 행렬을 "),t("strong",null,"특이행렬(Singular Matrix)"),Q(" 이라고 칭한다. 역행렬을 갖는 행렬은 가역행렬(Non Singular Matrix)이라고 부르기도 한다.")],-1),N1={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},J1={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"0"},xmlns:"http://www.w3.org/2000/svg",width:"3.929ex",height:"1.887ex",role:"img",focusable:"false",viewBox:"0 -833.9 1736.7 833.9","aria-hidden":"true"},z1=e('',1),G1=[z1],q1=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"A"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"−"),t("mn",null,"1")])])])],-1),F1=t("p",null,"역행렬을 본격적으로 이해해보기 전에, 헹렬식에 대해 알아보자.",-1),K1=t("h3",{id:"행렬식-determinant",tabindex:"-1"},[Q("행렬식 (determinant) "),t("a",{class:"header-anchor",href:"#행렬식-determinant","aria-label":'Permalink to "행렬식 (determinant)"'},"​")],-1),$1=t("p",null,"행렬식이란, 정방행렬에 하나의 수를 대응시키는 함수를 의미한다. 행렬을 통해 연립일차방정식의 해(크래머 공식)를 구하기 위해 고안되었다고 한다. 그 외에도 고윳값을 계산할 떄도 등장하는 용어다. 역행렬에서 행렬식에 대해 다루는 이유는, 행렬식을 통해 역행렬의 존재성을 판별하기 때문이다.",-1),W1=t("p",null,[Q("이차정사각행렬에서 기하학적으로 행렬식을 이해하면, 평면상에서 행렬의 열벡터를 표현했을때, 각각의 종점을 연장하여 평행사변형을 만들었을 때의 넓이가 행렬식의 값과 같다. "),t("img",{src:n,alt:"정사각행렬의 열벡터로 나타낸 평행사변형"})],-1),U1=t("p",null,"삼차정사각행렬에서의 행렬식의 값도 마찬가지로, 3차원 공간상에 표현된 평행육면체의 부피와 같다. 2X2 정방행렬에서의 행렬식은 다음과 같다",-1),Y1={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},t2={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-2.149ex"},xmlns:"http://www.w3.org/2000/svg",width:"20.556ex",height:"5.43ex",role:"img",focusable:"false",viewBox:"0 -1450 9085.7 2400","aria-hidden":"true"},Q2=e('',1),a2=[Q2],T2=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"d"),t("mi",null,"e"),t("mi",null,"t"),t("mrow",{"data-mjx-texclass":"INNER"},[t("mo",{"data-mjx-texclass":"OPEN"},"["),t("mtable",{columnspacing:"1em",rowspacing:"4pt"},[t("mtr",null,[t("mtd",null,[t("mi",null,"a")]),t("mtd",null,[t("mi",null,"b")])]),t("mtr",null,[t("mtd",null,[t("mi",null,"c")]),t("mtd",null,[t("mi",null,"d")])])]),t("mo",{"data-mjx-texclass":"CLOSE"},"]")]),t("mo",null,"="),t("mi",null,"a"),t("mi",null,"d"),t("mo",null,"−"),t("mi",null,"b"),t("mi",null,"c")])],-1),e2=e('

2X2 정방행렬에서의 행렬식 연산은 간단하다. 대각선을 이루는 원소들끼리 곱하고, 이들을 더하기만 하면 된다.

3X3 정방행렬에서의 행렬식 연산은 좀 복잡하다. 두 가지 방법으로 구해볼 수 있다.

전개 방식

행렬에서 마지막 열을 제외한 나머지 열을 마지막 열의 다음에 붙여준다

이 상태에서 다음 연산을 진행해준다.

3X3 행렬식 1

그 다음엔 반대로 구한다.

3X3 행렬식 2

마지막으로 앞서 구한 두 값을 뺀다.

3X3 행렬식 3

여인수와 소행렬 ​

앞서 짧게 소개한 전개 방식은 사실 여인수전개를 통한 계산을 간단히 한 것이다.

소행렬을 통한 계산을 위해선 소행렬과 여인수전개의 개념을 먼저 숙지해야 한다.

소행렬 (minor determinant)

소행렬이란, 특정 열과 행을 제거하고 만든 부분행렬에 대한 행렬식을 의미한다.

행렬에 제외되는 행, 열을 아래첨자로 표기하면 된다. 또는 소행렬에 절댓값 기호를 취해서 나타내는 방법도 있다.

위 식은 i행, j열의 원소들을 제하고 남은 부분에 대한 행렬식을 의미한다. 그림으로 나타내면 다음과 같다.

소행렬

여인수(cofactor)와 여인수전개

여인수전개는 여인수로 행렬식을 구하는 방법을 의미한다. 라플라스 전개라고 부르기도 한다.

여인수는 소행렬에 (-1)^(i+j)를 곱한 값을 의미한다. 식으로 나타내면 다음과 같다.

',14),l2={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},s2={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.666ex"},xmlns:"http://www.w3.org/2000/svg",width:"47.239ex",height:"2.8ex",role:"img",focusable:"false",viewBox:"0 -943.3 20879.4 1237.6","aria-hidden":"true"},o2=e('',1),n2=[o2],d2=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mi",null,"C"),t("mo",null,"="),t("mi",null,"C"),t("mi",null,"o"),t("mi",null,"f"),t("mi",null,"a"),t("mi",null,"t"),t("mi",null,"o"),t("mi",null,"r"),t("mo",null,","),t("msub",null,[t("mi",null,"A"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mi",null,"j")])]),t("mo",null,"="),t("mi",null,"S"),t("mi",null,"u"),t("mi",null,"b"),t("mi",null,"m"),t("mi",null,"a"),t("mi",null,"t"),t("mi",null,"r"),t("mi",null,"i"),t("mi",null,"x"),t("mspace",{linebreak:"newline"}),t("mspace",{linebreak:"newline"}),t("mi",null,"C"),t("mo",null,"="),t("mo",{stretchy:"false"},"("),t("mo",null,"−"),t("mn",null,"1"),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",{stretchy:"false"},"("),t("mi",null,"i"),t("mo",null,"+"),t("mi",null,"j"),t("mo",{stretchy:"false"},")")])]),t("msub",null,[t("mi",null,"A"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mi",null,"i"),t("mi",null,"j")])])])],-1),m2=t("p",null,"여인수 전개에서는 어떠한 행, 열에 대해 여인수 전개를 진행해도 동일한 값이 도출된다. 따라서, 한 행이나 열이 모두 동일한 값이거나 0이 많이 포함된 경우, 이 행/열에 대해 여인수 전개를 진행하면 매우 효율적으로 계산을 진행할 수 있다.",-1),r2=t("p",null,"여인수전개는 4x4, 5x5 등 모든 차원의 정방행렬에 대해 적용될 수 있다.",-1),i2=t("p",null,"다음은 3X3 행렬에서 여인수전개로 행렬식 값을 구한 과정을 나타낸 그림이다.",-1),h2=t("p",null,[t("img",{src:h,alt:"여인수 전개"})],-1),p2=t("p",null,"위 그림대로 식들을 전개하면 다음의 식을 얻을 수 있다.",-1),c2={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},g2={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"63.037ex",height:"2.7ex",role:"img",focusable:"false",viewBox:"0 -943.3 27862.3 1193.3","aria-hidden":"true"},H2=e('',1),u2=[H2],w2=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 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0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("msub",null,[t("mi",null,"a"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"21")])]),t("mo",null,"∗"),t("mo",{stretchy:"false"},"("),t("mo",null,"−"),t("mn",null,"1"),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",{stretchy:"false"},"("),t("mn",null,"2"),t("mo",null,"+"),t("mn",null,"1"),t("mo",{stretchy:"false"},")")])]),t("mo",null,"⋅"),t("msub",null,[t("mi",null,"A"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"21")])]),t("mo",null,"+"),t("msub",null,[t("mi",null,"a"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"22")])]),t("mo",null,"∗"),t("mo",{stretchy:"false"},"("),t("mo",null,"−"),t("mn",null,"1"),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",{stretchy:"false"},"("),t("mn",null,"2"),t("mo",null,"+"),t("mn",null,"2"),t("mo",{stretchy:"false"},")")])]),t("mo",null,"⋅"),t("msub",null,[t("mi",null,"A"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"22")])]),t("mo",null,"+"),t("msub",null,[t("mi",null,"a"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"23")])]),t("mo",null,"∗"),t("mo",{stretchy:"false"},"("),t("mo",null,"−"),t("mn",null,"1"),t("msup",null,[t("mo",{stretchy:"false"},")"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",{stretchy:"false"},"("),t("mn",null,"2"),t("mo",null,"+"),t("mn",null,"3"),t("mo",{stretchy:"false"},")")])]),t("mo",null,"⋅"),t("msub",null,[t("mi",null,"A"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mn",null,"23")])])])],-1),x2=t("p",null,"실제 값을 대입하면 다음의 결과를 얻을 수 있다.",-1),L2={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},_2={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"78.061ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 34503.1 1000","aria-hidden":"true"},y2=e('',1),f2=[y2],M2=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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코드상으로 구현하면, 5X5 에서 4X4, 3X3 ... 으로 점차 줄어들어 쉽게 계산할 수 있는 2X2의 소행렬로 나눠 계산한 후, 다른 계산 결과와 합쳐 나가는 방식을 취할 것이다. 이처럼 자기 자신을 더 작게 나누어 계산 가능한 사이즈로 나눠 계산한 후 원래의 상태로 거슬러 올라가며 합쳐나가는 알고리즘은 재귀적 프로그래밍으로 구현했을 때 효과적이다. 하지만 대체로 텐서 연산 관련 라이브러리에 잘 구현되어 있다.

파이썬 상에서 행렬식은 구하고자 한다면, 복잡한 구현 없이 다음의 코드로 쉽게 구할 수 있다.

X = np.array([[1,2,4],[2,-1,3],[0,5,1]])
-np.linalg.det(X)

행렬식과 역행렬 ​

행렬식을 알아본 이유는, 행렬식을 통해 정방행렬에 역행렬이 존재하는지를 확인할 수 있기 때문이다.

`,5),V2=t("strong",null,"역행렬이 존재하기 위해선, 행렬식의 값이 0이 아니어야 한다.",-1),Z2={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},b2={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.566ex"},xmlns:"http://www.w3.org/2000/svg",width:"10.653ex",height:"2.262ex",role:"img",focusable:"false",viewBox:"0 -750 4708.6 1000","aria-hidden":"true"},v2=e('',1),D2=[v2],j2=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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직교행렬과 직교행렬의 곱은 직교행렬이다. 이는 직교행렬이 각도와 길이, 내적을 보존하는 행렬이기 때문이다.

Ref ​

[선형대수학] Determinant / 행렬식

행렬의 성분, 두 행렬이 서로 같을 조건

행렬식 이 영상만 보면 기본 끝! | 행렬식의 성질 | 기본행연산과 행렬식

orthogonal matrix(직교행렬) 과 matrix of orthonormal columns

',6);function k3(V3,Z3,b3,v3,D3,j3){return T(),a("div",null,[c,g,t("p",null,[Q("대칭 행렬은 자신의 전치 행렬이 원래의 자기 자신과 같은 행렬이다. 즉, "),t("mjx-container",H,[(T(),a("svg",u,x)),L]),Q(" 인 행렬을 의미한다.")]),t("p",null,[Q("대칭행렬을 이루기 위해선 원소의 row index와 column index가 반대로 바뀌어도 동일한 원소를 가져야 한다. "),t("mjx-container",_,[(T(),a("svg",y,M)),k]),Q(" 라는 것이다.")]),V,t("p",null,[Q("여기서, "),t("mjx-container",Z,[(T(),a("svg",b,D)),j]),Q(" 인 원소들을 "),C,Q(" 이라고 한다. 대칭행렬은 대각성분을 기준으로 대칭을 이룬다.")]),A,S,R,O,t("p",null,[Q("단위행렬은 "),t("mjx-container",P,[(T(),a("svg",E,B)),X]),Q("으로 나타낸다. n은 행(열)의 개수를 의미한다. 벡터에 단위 연산을 곱하는 연산을 수식으로 나타내면 다음과 같다")]),t("mjx-container",N,[(T(),a("svg",J,G)),q]),F,K,$,W,U,Y,t("p",null,[Q("대각성분벡터(주대각선)로 대각행렬을 표기하는 방법은 다음과 같다. 이때 "),t("mjx-container",t1,[(T(),a("svg",Q1,T1)),e1]),Q("는 주대각선을 이루는 대각성분벡터이다.")]),t("mjx-container",l1,[(T(),a("svg",s1,n1)),d1]),t("p",null,[Q("대각행렬의 대각성분을 이루는 벡터("),t("mjx-container",m1,[(T(),a("svg",r1,h1)),p1]),Q(")와 벡터의 곱셈연산은 아마다르곱("),t("mjx-container",c1,[(T(),a("svg",g1,u1)),w1]),Q(")과 같다.")]),t("mjx-container",x1,[(T(),a("svg",L1,y1)),f1]),M1,t("p",null,[Q("역행렬은 같은 꼴의 정방행렬 "),t("mjx-container",k1,[(T(),a("svg",V1,b1)),v1]),Q("와 단위행렬 "),t("mjx-container",D1,[(T(),a("svg",j1,A1)),S1]),Q("에 대해 "),t("mjx-container",R1,[(T(),a("svg",O1,E1)),I1]),Q("를 만족시키는 행렬을 의미한다. 일반적으로 행렬곱은 교환법칙이 성립하지 않아 곱셉 순서를 바꾸면 그 결과가 달라지나, 역행렬의 경우 곱셈의 순서를 바꾸어도 그 결과가 단위행렬로 항상 동일하다.")]),B1,X1,t("p",null,[Q("역행렬에 대한 표기는 "),t("mjx-container",N1,[(T(),a("svg",J1,G1)),q1]),Q("로 한다. 분수의 표기와 동일하다. 단위행렬과 행렬의 곱이 스칼라 1을 행렬에 곱한 결과와 비슷하다는 점에서 분수가 연상되기도 한다.")]),F1,K1,$1,W1,U1,t("mjx-container",Y1,[(T(),a("svg",t2,a2)),T2]),e2,t("mjx-container",l2,[(T(),a("svg",s2,n2)),d2]),m2,r2,i2,h2,p2,t("mjx-container",c2,[(T(),a("svg",g2,u2)),w2]),x2,t("mjx-container",L2,[(T(),a("svg",_2,f2)),M2]),k2,t("p",null,[V2,Q(" 즉, "),t("mjx-container",Z2,[(T(),a("svg",b2,D2)),j2]),Q("이 역행렬의 성립조건이다. 앞서 언급했듯이, 행렬식의 값은 벡터의 종점을 벡터의 크기만큼 연장시킨 평행사변형의 넓이이다. 행렬값이 0이라는 것은 다시 말하면, 두 벡터가 한 직선 상에 놓인 것이나 마찬가지이다.")]),C2,t("mjx-container",A2,[(T(),a("svg",S2,O2)),P2]),E2,I2,B2,t("p",null,[Q("또한, 직교행렬의 전치행렬과 직교 행렬의 곱은 단위행렬이다. 즉, "),t("mjx-container",X2,[(T(),a("svg",N2,z2)),G2]),Q(" 이다. 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833.9","aria-hidden":"true"},z1=e("",1),G1=[z1],q1=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"A"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"−"),t("mn",null,"1")])])])],-1),F1=t("p",null,"역행렬을 본격적으로 이해해보기 전에, 헹렬식에 대해 알아보자.",-1),K1=t("h3",{id:"행렬식-determinant",tabindex:"-1"},[Q("행렬식 (determinant) "),t("a",{class:"header-anchor",href:"#행렬식-determinant","aria-label":'Permalink to "행렬식 (determinant)"'},"​")],-1),$1=t("p",null,"행렬식이란, 정방행렬에 하나의 수를 대응시키는 함수를 의미한다. 행렬을 통해 연립일차방정식의 해(크래머 공식)를 구하기 위해 고안되었다고 한다. 그 외에도 고윳값을 계산할 떄도 등장하는 용어다. 역행렬에서 행렬식에 대해 다루는 이유는, 행렬식을 통해 역행렬의 존재성을 판별하기 때문이다.",-1),W1=t("p",null,[Q("이차정사각행렬에서 기하학적으로 행렬식을 이해하면, 평면상에서 행렬의 열벡터를 표현했을때, 각각의 종점을 연장하여 평행사변형을 만들었을 때의 넓이가 행렬식의 값과 같다. 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0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"Q"),t("mi",null,"T")]),t("mi",null,"Q"),t("mo",null,"="),t("mi",null,"I")])],-1),U2=t("strong",null,"역행렬이 곧 전치행렬",-1),Y2={class:"MathJax",jax:"SVG",style:{direction:"ltr",position:"relative"}},t3={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.439ex"},xmlns:"http://www.w3.org/2000/svg",width:"10.143ex",height:"2.343ex",role:"img",focusable:"false",viewBox:"0 -841.7 4483 1035.7","aria-hidden":"true"},Q3=e("",1),a3=[Q3],T3=t("mjx-assistive-mml",{unselectable:"on",display:"inline",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",width:"auto",overflow:"hidden"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML"},[t("msup",null,[t("mi",null,"Q"),t("mi",null,"T")]),t("mo",null,"="),t("msup",null,[t("mi",null,"Q"),t("mrow",{"data-mjx-texclass":"ORD"},[t("mo",null,"−"),t("mn",null,"1")])])])],-1),e3=t("p",null,"직교행렬과 벡터의 곱의 크기는 벡터의 크기와 같다. 이는 아래의 식으로 증명 할 수 있다.",-1),l3={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 0",position:"relative"}},s3={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.564ex"},xmlns:"http://www.w3.org/2000/svg",width:"12.033ex",height:"2.26ex",role:"img",focusable:"false",viewBox:"0 -749.5 5318.6 999","aria-hidden":"true"},o3=e("",1),n3=[o3],d3=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 0px",border:"0px",display:"block",overflow:"hidden",width:"100%"}},[t("math",{xmlns:"http://www.w3.org/1998/Math/MathML",display:"block"},[t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mi",null,"Q"),t("mi",null,"v"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mo",null,"="),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mi",null,"v"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|"),t("mo",{"data-mjx-texclass":"ORD",stretchy:"false"},"|")])],-1),m3={class:"MathJax",jax:"SVG",display:"true",style:{direction:"ltr",display:"block","text-align":"center",margin:"1em 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0",position:"relative"}},L3={style:{overflow:"visible","min-height":"1px","min-width":"1px","vertical-align":"-0.564ex"},xmlns:"http://www.w3.org/2000/svg",width:"10.139ex",height:"2.582ex",role:"img",focusable:"false",viewBox:"0 -891.7 4481.4 1141.2","aria-hidden":"true"},_3=e("",1),y3=[_3],f3=t("mjx-assistive-mml",{unselectable:"on",display:"block",style:{top:"0px",left:"0px",clip:"rect(1px, 1px, 1px, 1px)","-webkit-touch-callout":"none","-webkit-user-select":"none","-khtml-user-select":"none","-moz-user-select":"none","-ms-user-select":"none","user-select":"none",position:"absolute",padding:"1px 0px 0px 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"),t("mjx-container",_,[(T(),a("svg",y,M)),k]),Q(" 라는 것이다.")]),V,t("p",null,[Q("여기서, "),t("mjx-container",Z,[(T(),a("svg",b,D)),j]),Q(" 인 원소들을 "),C,Q(" 이라고 한다. 대칭행렬은 대각성분을 기준으로 대칭을 이룬다.")]),A,S,R,O,t("p",null,[Q("단위행렬은 "),t("mjx-container",P,[(T(),a("svg",E,B)),X]),Q("으로 나타낸다. n은 행(열)의 개수를 의미한다. 벡터에 단위 연산을 곱하는 연산을 수식으로 나타내면 다음과 같다")]),t("mjx-container",N,[(T(),a("svg",J,G)),q]),F,K,$,W,U,Y,t("p",null,[Q("대각성분벡터(주대각선)로 대각행렬을 표기하는 방법은 다음과 같다. 이때 "),t("mjx-container",t1,[(T(),a("svg",Q1,T1)),e1]),Q("는 주대각선을 이루는 대각성분벡터이다.")]),t("mjx-container",l1,[(T(),a("svg",s1,n1)),d1]),t("p",null,[Q("대각행렬의 대각성분을 이루는 벡터("),t("mjx-container",m1,[(T(),a("svg",r1,h1)),p1]),Q(")와 벡터의 곱셈연산은 아마다르곱("),t("mjx-container",c1,[(T(),a("svg",g1,u1)),w1]),Q(")과 같다.")]),t("mjx-container",x1,[(T(),a("svg",L1,y1)),f1]),M1,t("p",null,[Q("역행렬은 같은 꼴의 정방행렬 "),t("mjx-container",k1,[(T(),a("svg",V1,b1)),v1]),Q("와 단위행렬 "),t("mjx-container",D1,[(T(),a("svg",j1,A1)),S1]),Q("에 대해 "),t("mjx-container",R1,[(T(),a("svg",O1,E1)),I1]),Q("를 만족시키는 행렬을 의미한다. 일반적으로 행렬곱은 교환법칙이 성립하지 않아 곱셉 순서를 바꾸면 그 결과가 달라지나, 역행렬의 경우 곱셈의 순서를 바꾸어도 그 결과가 단위행렬로 항상 동일하다.")]),B1,X1,t("p",null,[Q("역행렬에 대한 표기는 "),t("mjx-container",N1,[(T(),a("svg",J1,G1)),q1]),Q("로 한다. 분수의 표기와 동일하다. 단위행렬과 행렬의 곱이 스칼라 1을 행렬에 곱한 결과와 비슷하다는 점에서 분수가 연상되기도 한다.")]),F1,K1,$1,W1,U1,t("mjx-container",Y1,[(T(),a("svg",t2,a2)),T2]),e2,t("mjx-container",l2,[(T(),a("svg",s2,n2)),d2]),m2,r2,i2,h2,p2,t("mjx-container",c2,[(T(),a("svg",g2,u2)),w2]),x2,t("mjx-container",L2,[(T(),a("svg",_2,f2)),M2]),k2,t("p",null,[V2,Q(" 즉, "),t("mjx-container",Z2,[(T(),a("svg",b2,D2)),j2]),Q("이 역행렬의 성립조건이다. 앞서 언급했듯이, 행렬식의 값은 벡터의 종점을 벡터의 크기만큼 연장시킨 평행사변형의 넓이이다. 행렬값이 0이라는 것은 다시 말하면, 두 벡터가 한 직선 상에 놓인 것이나 마찬가지이다.")]),C2,t("mjx-container",A2,[(T(),a("svg",S2,O2)),P2]),E2,I2,B2,t("p",null,[Q("또한, 직교행렬의 전치행렬과 직교 행렬의 곱은 단위행렬이다. 즉, "),t("mjx-container",X2,[(T(),a("svg",N2,z2)),G2]),Q(" 이다. 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diff --git a/docs/.vitepress/dist/assets/contents_VITEPRESS_git-deploy.md.BsMrOoNL.js b/docs/.vitepress/dist/assets/contents_VITEPRESS_git-deploy.md.B5KVrLUN.js similarity index 98% rename from docs/.vitepress/dist/assets/contents_VITEPRESS_git-deploy.md.BsMrOoNL.js rename to docs/.vitepress/dist/assets/contents_VITEPRESS_git-deploy.md.B5KVrLUN.js index a33fe1a..081bf98 100644 --- a/docs/.vitepress/dist/assets/contents_VITEPRESS_git-deploy.md.BsMrOoNL.js +++ b/docs/.vitepress/dist/assets/contents_VITEPRESS_git-deploy.md.B5KVrLUN.js @@ -1,4 +1,4 @@ -import{_ as s,c as i,o as a,a5 as n}from"./chunks/framework.BuWuHeYF.js";const e="/assets/vitepress-deploy-action.BTTKGiaD.png",y=JSON.parse('{"title":"Vitepress로 Github Page 디플로이 하기","description":"Vitepress로 Github page 디플로이 하는 방법","frontmatter":{"title":"Vitepress로 Github Page 디플로이 하기","description":"Vitepress로 Github page 디플로이 하는 방법","keywords":"\\"vitepress\\", \\"github\\", \\"deploy\\", \\"깃허브\\", \\"디플로이\\"","url":"VITEPRESS/git-deploy.html","head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsVITEPRESS/git-deploy.html\\"\\n },\\n \\"name\\":\\"Vitepress로 Github Page 디플로이 하기\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/VITEPRESS/git-deploy.html\\",\\n \\"headline\\":\\"Vitepress로 Github page 디플로이 하는 방법\\",\\n \\"description\\":\\"Vitepress로 Github page 디플로이 하는 방법\\",\\n \\"keywords\\" : [\\"vitepress\\", \\"github\\", \\"deploy\\", \\"깃허브\\", \\"디플로이\\"],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title","content":"Vitepress로 Github Page 디플로이 하기"}],["meta",{"property":"og:description","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"og:url","content":"VITEPRESS/git-deploy.html"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"twitter:title","content":"Vitepress로 Github Page 디플로이 하기"}],["meta",{"property":"twitter:description","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"twitter:image","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name","content":"Vitepress로 Github Page 디플로이 하기"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/VITEPRESS/git-deploy.html"}],["meta",{"property":"description","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"keywords","content":""vitepress", "github", "deploy", "깃허브", "디플로이""}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/VITEPRESS/git-deploy.md","filePath":"contents/VITEPRESS/git-deploy.md","lastUpdated":1711338894000}'),p={name:"contents/VITEPRESS/git-deploy.md"},t=n(`

Vitepress로 깃허브 페이지 디플로이 하기 ​

💡Vitepress 공식문서에서 잘 설명하고 있다. 다른 서비스로 호스팅하고 싶으면 공식문서를 잘 살펴보자

이미 디플로이를 위한 깃허브 페이지용 레포지터리가 존재한다는 전제 하에 다음 과정을 수행하면 된다. 깃허브 페이지 레포지터리 생성 방법은 다음의 링크를 참고하면 된다.

STEP 1. 폴더 생성 ​

docs 폴더와 동일한 위치에 .github/workflows 폴더를 만든다. workflows 폴더 안에 deploy.yml 파일을 하나 만든다.

bash
├── .github
+import{_ as s,c as i,o as a,a5 as n}from"./chunks/framework.BuWuHeYF.js";const e="/assets/vitepress-deploy-action.BTTKGiaD.png",y=JSON.parse('{"title":"Vitepress로 Github Page 디플로이 하기","description":"Vitepress로 Github page 디플로이 하는 방법","frontmatter":{"title":"Vitepress로 Github Page 디플로이 하기","description":"Vitepress로 Github page 디플로이 하는 방법","keywords":"\\"vitepress\\", \\"github\\", \\"deploy\\", \\"깃허브\\", \\"디플로이\\"","url":"VITEPRESS/git-deploy.html","head":[["script",{"type":"application/ld+json"},"{\\n  \\"@context\\":\\"http://schema.org\\",\\n  \\"@type\\":\\"BlogPosting\\",\\n  \\"mainEntityOfPage\\" : {\\n    \\"@type\\" : \\"WebPage\\",\\n    \\"@id\\" : \\"https://an-jieun.github.io/contentsVITEPRESS/git-deploy.html\\"\\n  },\\n  \\"name\\":\\"Vitepress로 Github Page 디플로이 하기\\",\\n  \\"url\\" : \\"https://an-jieun.github.io/contents/VITEPRESS/git-deploy.html\\",\\n  \\"headline\\":\\"Vitepress로 Github page 디플로이 하는 방법\\",\\n  \\"description\\":\\"Vitepress로 Github page 디플로이 하는 방법\\",\\n  \\"keywords\\" : [\\"vitepress\\", \\"github\\", \\"deploy\\", \\"깃허브\\", \\"디플로이\\"],\\n  \\"inLanguage\\":\\"ko\\",\\n  \\"author\\" : {\\n    \\"@type\\" : \\"Person\\",\\n    \\"name\\" : \\"Jieun\\",\\n    \\"email\\" : \\"aje20010827@gmail.com\\"\\n    }\\n  },\\n}"],["meta",{"property":"og:title","content":"Vitepress로 Github Page 디플로이 하기"}],["meta",{"property":"og:description","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"og:url","content":"VITEPRESS/git-deploy.html"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"twitter:title","content":"Vitepress로 Github Page 디플로이 하기"}],["meta",{"property":"twitter:description","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"twitter:image"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name","content":"Vitepress로 Github Page 디플로이 하기"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/VITEPRESS/git-deploy.html"}],["meta",{"property":"description","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"keywords","content":""vitepress", "github", "deploy", "깃허브", "디플로이""}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/VITEPRESS/git-deploy.md","filePath":"contents/VITEPRESS/git-deploy.md","lastUpdated":1711338894000}'),p={name:"contents/VITEPRESS/git-deploy.md"},l=n(`
Vitepress로 깃허브 페이지 디플로이 하기 ​
💡Vitepress 공식문서에서 잘 설명하고 있다. 다른 서비스로 호스팅하고 싶으면 공식문서를 잘 살펴보자
이미 디플로이를 위한 깃허브 페이지용 레포지터리가 존재한다는 전제 하에 다음 과정을 수행하면 된다. 깃허브 페이지 레포지터리 생성 방법은 다음의 링크를 참고하면 된다.
STEP 1. 폴더 생성 ​
docs 폴더와 동일한 위치에 .github/workflows 폴더를 만든다. workflows 폴더 안에 deploy.yml 파일을 하나 만든다.
bash
├── .github
 │   └── workflows
 │       └── deploy.yml
 └── docs
STEP 2. deploy.yml 파일 생성 ​
deploy.yml파일을 작성한다. 아래 내용을 복붙하고 주석처리된 부분만 자신에게 맞게 수정하면 된다.
yaml

@@ -58,4 +58,4 @@ import{_ as s,c as i,o as a,a5 as n}from"./chunks/framework.BuWuHeYF.js";const e
     steps:
       - name: Deploy to GitHub Pages
         id: deployment
-        uses: actions/deploy-pages@v4
STEP 3. 깃허브 Actions 탭에서 디플로이 되고 있는지 확인 ​
이제 디플로이용 브랜치에서 npm run docs:build를 수행한 후 push 하면 디플로이가 자동으로 실행된다. 디플로이 진행상황은 깃허브 action 탭에서 확인하면 된다. 디플로이가 완료되면 아래 사진처럼 초록색 체크 아이콘이 뜬다.
action tab
',12),l=[t];function h(k,r,E,d,c,o){return a(),i("div",null,l)}const u=s(p,[["render",h]]);export{y as __pageData,u as default};
+        uses: actions/deploy-pages@v4

STEP 3. 깃허브 Actions 탭에서 디플로이 되고 있는지 확인 ​

이제 디플로이용 브랜치에서 npm run docs:build를 수행한 후 push 하면 디플로이가 자동으로 실행된다. 디플로이 진행상황은 깃허브 action 탭에서 확인하면 된다. 디플로이가 완료되면 아래 사진처럼 초록색 체크 아이콘이 뜬다.

action tab

',12),t=[l];function h(k,r,E,d,c,o){return a(),i("div",null,t)}const u=s(p,[["render",h]]);export{y as __pageData,u as default}; diff --git a/docs/.vitepress/dist/assets/contents_VITEPRESS_git-deploy.md.BsMrOoNL.lean.js b/docs/.vitepress/dist/assets/contents_VITEPRESS_git-deploy.md.B5KVrLUN.lean.js similarity index 92% rename from docs/.vitepress/dist/assets/contents_VITEPRESS_git-deploy.md.BsMrOoNL.lean.js rename to docs/.vitepress/dist/assets/contents_VITEPRESS_git-deploy.md.B5KVrLUN.lean.js index a74ea22..5e77a0d 100644 --- a/docs/.vitepress/dist/assets/contents_VITEPRESS_git-deploy.md.BsMrOoNL.lean.js +++ b/docs/.vitepress/dist/assets/contents_VITEPRESS_git-deploy.md.B5KVrLUN.lean.js @@ -1 +1 @@ -import{_ as s,c as i,o as a,a5 as n}from"./chunks/framework.BuWuHeYF.js";const e="/assets/vitepress-deploy-action.BTTKGiaD.png",y=JSON.parse('{"title":"Vitepress로 Github Page 디플로이 하기","description":"Vitepress로 Github page 디플로이 하는 방법","frontmatter":{"title":"Vitepress로 Github Page 디플로이 하기","description":"Vitepress로 Github page 디플로이 하는 방법","keywords":"\\"vitepress\\", \\"github\\", \\"deploy\\", \\"깃허브\\", \\"디플로이\\"","url":"VITEPRESS/git-deploy.html","head":[["script",{"type":"application/ld+json"},"{\\n \\"@context\\":\\"http://schema.org\\",\\n \\"@type\\":\\"BlogPosting\\",\\n \\"mainEntityOfPage\\" : {\\n \\"@type\\" : \\"WebPage\\",\\n \\"@id\\" : \\"https://an-jieun.github.io/contentsVITEPRESS/git-deploy.html\\"\\n },\\n \\"name\\":\\"Vitepress로 Github Page 디플로이 하기\\",\\n \\"url\\" : \\"https://an-jieun.github.io/contents/VITEPRESS/git-deploy.html\\",\\n \\"headline\\":\\"Vitepress로 Github page 디플로이 하는 방법\\",\\n \\"description\\":\\"Vitepress로 Github page 디플로이 하는 방법\\",\\n \\"keywords\\" : [\\"vitepress\\", \\"github\\", \\"deploy\\", \\"깃허브\\", \\"디플로이\\"],\\n \\"inLanguage\\":\\"ko\\",\\n \\"author\\" : {\\n \\"@type\\" : \\"Person\\",\\n \\"name\\" : \\"Jieun\\",\\n \\"email\\" : \\"aje20010827@gmail.com\\"\\n }\\n },\\n}"],["meta",{"property":"og:title","content":"Vitepress로 Github Page 디플로이 하기"}],["meta",{"property":"og:description","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"og:url","content":"VITEPRESS/git-deploy.html"}],["meta",{"property":"og:type","content":"website"}],["meta",{"property":"og:site_name","content":"전자두뇌만들기"}],["meta",{"property":"og:locale","content":"ko_KR"}],["meta",{"property":"twitter:card","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"twitter:title","content":"Vitepress로 Github Page 디플로이 하기"}],["meta",{"property":"twitter:description","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"twitter:image","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"@context","content":"http://schema.org"}],["meta",{"property":"@type","content":"TechArticle"}],["meta",{"property":"name","content":"Vitepress로 Github Page 디플로이 하기"}],["meta",{"property":"url","content":"https://an-jieun.github.io/contents/VITEPRESS/git-deploy.html"}],["meta",{"property":"description","content":"Vitepress로 Github page 디플로이 하는 방법"}],["meta",{"property":"keywords","content":""vitepress", "github", "deploy", "깃허브", "디플로이""}],["meta",{"property":"version","content":"1.0"}],["meta",{"property":"inLanguage","content":"ko"}],["meta",{"property":"technicalAudience","content":"developer, DBA, Web Developer"}],["meta",{"property":"proficiencyLevel","content":"beginner"}],["meta",{"property":"author","content":"Jieun"}],["meta",{"property":"dependencies","content":"Python"}]]},"headers":[],"relativePath":"contents/VITEPRESS/git-deploy.md","filePath":"contents/VITEPRESS/git-deploy.md","lastUpdated":1711338894000}'),p={name:"contents/VITEPRESS/git-deploy.md"},t=n("",12),l=[t];function h(k,r,E,d,c,o){return a(),i("div",null,l)}const u=s(p,[["render",h]]);export{y as __pageData,u as default}; 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diff --git a/docs/.vitepress/dist/contents/KG/kg-main.html b/docs/.vitepress/dist/contents/KG/kg-main.html index 71c91fa..b393bb3 100644 --- a/docs/.vitepress/dist/contents/KG/kg-main.html +++ b/docs/.vitepress/dist/contents/KG/kg-main.html @@ -63,7 +63,7 @@
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Knowledge Graph ​

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[지식그래프] Knowledge Graph for Beginners 정리 ​

Materials

  • Udemy 강의 : Knowledge Graph for Beginners : A concise introduction to knowledge graphs for complete beginners, (Tish Chungoora)

Knowledge Graph for Beginners

지식과 그래프 ​

💡 지식그래프란?

명시적으로 정의된 관계로 연결된 사실(facts)의 네트워크로, 관계를 통해 새로운 지식이 유추 될 수 있다. 지식그래프는 ‘온톨로지’로 알려진 기반 구조나 스키마에 따라 구축된다.

  • 사실의 네트워크

    • 지식그래프에서의 지식의 기본 단위는 사실과, 해당 사실과 관계된 사실, 그리고 두 사실 간의 관계로 표현된다.

    • 이러한 아이디어는 그래프 노드(정점)와 노드(정점)사이를 간선으로 표현하는 그래프 아이디어에서 비롯되었다.

      • 지식그래프와 언어모델

        지식그래프가 기존의 그래프 이론에서 발전한 만큼, 지식그래프도 node2vec, attention을 거쳐 발전한 언어 모델에 활용될 수 있으며, 릴레이션으로 단어 형태소 간의 자세한 관계 표현이 가능하다는 점에서 설명 가능한 언어모델을 만들 수 있을 것이라는 기대도 있었다.

    • 사실 네트워크(network of facts)의 기본단위는 ‘사실, 사실 관계에 있는 사실, 관계’(fact—relation—>fact)로 구성되며, triple(트리플) 로 불린다.

  • 명시적으로 정의된 관계(relation)

    • 사실과 사실의 연결 짓는 요소로, 레이블을 가지며 대부분의 경우 방향성을 갖는다. 속성을 가질 수도 있다. 사람은 관계의 레이블을 바탕으로 사실과 사실간의 관계를 해석할 수 있다.
    • 클래스는 엔티티가 가질 수 있는 관계를 제한하여 무결성 제약 조건을 충족시킬 수 있다.

  • 기반 구조

    • 클래스(Class)

      강의 교안에서는 ‘Type of kind’ 로 정의하고 있다. 그대로 번역하면 ‘종류의 종류(…)’ 인데, 영어사전에서는 type과 kind를 약간 다르게 정의하고 있다.

      우선 kind는 ‘a type of thing or person’이고, type은 ‘group of people or things that have similar qualities’ 이다.

      즉, type은 ‘동질성을 갖는 객체들의 그룹’이며 kind는 특정 개체가 속한 동질성을 갖는 객체 그룹(type)을 의미한다.

      종합하여 해석하면 Class란, ‘어떤 집합에서, 동일한 속성을 공유한 객체들의 부분집합’이라고 할 수 있다. 동일한 concept을 갖는 객체 집합으로 생각하면 편하다.

      클래스 내부의 개별 객체를 이르는 단어로는 보통 ‘entity’를 사용한다. 클래스 객체라고도 부른다.

    • 텍소노미(taxonomy ; 분류체계)

      텍소노미에 대한 정의는 ‘클래스를 정렬하는 규칙’이다. 번역된 어휘는 ‘분류 체계’이다. 현실 세계의 객체를 특정한 클래스로 분류하는 체계라고 이해하면 쉽다.

      텍소노미에서 발전한게 계통학인 것 같다..(확실하진 않다)

      십진분류를 생각하면 쉬운데, 분류체계는 일반적으로 계층구조를 가지며, 트리로 표현되기도 한다.

      • (+) Folksonomy

        대중에서 형성된 분류 체계를 의미한다. 즉, 컨텐츠의 주체가 생각하는 분류가 아닌 사용자 집단이 생각하는 분류이다. 대표적인 폭소노미 예시로 인스타그램 태깅을 들 수 있다. 인스타그램에서 게시물의 키워드를 나타내기도 하는 해시태그는 접근점으로 사용되며 사용자가 마음대로 생성할 수 있다.

    • 온톨로지

      존재하는 사물과 사물 간의 관계, 여러 개념을 컴퓨터가 처리할 수 있는 형태로 표현한 것.

      *온톨로지는 사물의 본질, 존재의 근본 원리를 사유나 직관에 의해 탐구하는 형이상학의 한 분야인 존재론을 기반으로 실재에 대한 정확한 이론을 추구하는 철학에서 유래되었다. 말 그대로, 온톨로지란 현실세계의 객체가 이루는 관계를 정합적으로 기술하는 도식(schema)이다.

      온톨로지는 클래스가 가질 수 있는 관계의 종류, 그리고 관계가 갖는 속성에 대한 정의를 포함한다.

    • 텍소노미와 온톨로지

      텍소노미는 클래스에 대한 분류 체계고, 온톨로지는 클래스에 따라 클래스 객체가 갖는 의미론적 연결 관계에 대한 규칙이다. 둘 다 어떤 체계에 대한 것이라 하니 서로 배타적인 관계에 있는 것 같기도 하지만 텍소노미와 온톨로지는 상호 유기적이다.

      십진분류표로 이 책이 어떤 분류에 속하는지 정할 수 있지만, 이 책과 다른 책의 관계에 대해선 정의할 수 없다는 것을 생각하면 쉽다.

      • 예를 들어 십진분류체계에서 ‘순수과학’이라는 주제(클래스)는 ‘오직 자연현상 만을 다루며 공학적 활용이 일부 포함된 경우도 해당 주제에 분류함’이라는 분류 규칙이 있으며,

        십진분류체계를 준용하는 ‘문헌 온톨로지’라는 가상의 온톨로지에서 ‘ ‘순수과학’ 주제가 철학에 영향을 받았’다는 관계 규칙이 있다고 하자.

        ‘철학’ 주제에는 A 도서가, ‘순수과학’ 주제에는 ‘B’도서가 분류되었다고 하자. ‘B’도서는 과학적 이론 설명과 함께 공학적 활용 분야도 함께 소개하고 있다. ‘문헌 온톨로지’ 상에서는 A 도서가 B 도서에 영향을 주었다고 나타난다.

        만약 십진분류체계가 수정되어 ‘순수과학’은 ‘오직 과학적 이론에 대한 내용만 다루며 공학적 활용에 대한 내용이 포함된 경우라도 ‘기술과학’으로 분류함.’ 으로 분류체계가 바뀐다면, A도서의 클래스는 바뀌며, 이에 따라 A도서와 B도서의 관계 역시 사라지게 된다.

      이렇듯, 텍소노미에 따라 개별 객체를 특정 클래스로 정의한다면, 온톨로지에서는 이를 바탕으로 클래스의 관계 규칙, 그리고 관계의 속성을 정의하게 되므로, 텍소노미에서의 체계를 수정한다면 이것이 온톨로지에도 영향을 미친다고 할 수 있다.

  • 새로운 지식의 유추

    kg-example

    객체들의 관계를 바탕으로 아직 정의되지 않은 관계를 유추하는 것을 의미한다. 쿼리를 통한 방법, 인공지능을 활용한 방법, 머신러닝을 활용하는 방법 등이 있다.

    • membership 자동 계산
    • Path Querying
    • Entity Resolution

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Neo4j 사용해보기 ​

Cypher ​

Cypher란 neo4j만의 그래프 쿼리 언어이다. 노드와 노드간의 관계를 표현한다. 기본 문법은 다음과 같다.

(nodes)-[:CONNECT_TO]→(otherNodes)

**()**는 노드를, **[:]**는 relation을 나타낸다.

Cypher로 Query하기 ​

cypher를 통한 질의는 훨씬 간편하고 정확하다. 테이블의 join 없이도 cypher를 활용한 질의로 단 두 줄 만에 쿼리할 수 있다.

예르 들어, 영화 matrix에 출연한 배우를 찾는다고 하면

MATCH (actor:Actor)-[:ACTED_IN]->(movie:Movie {title: 'The Matrix'})
 RETURN actor.name

두 쿼리 문으로 찾을 수 있다.

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온톨로지의 설계 과정 ​

1. 요구분석과 기초적인 시각화 ​

온톨로지는 요구사항에 맞춰 도메인의 클래스, 인스턴스, 논리적 포함관계(substances), 관계(association)을 정의한다. 이들에 대해 나타내는 용어를 '단어(term)'이라고 하며, 보통 설계 초기에는 단어를 정의하고 이들의 실질적 관계를 엑셀 시트 등의 테이블 형태 문서에 정리한다.

단어에 대한 정의가 완료되면, 이들의 관계를 그래프 구조로 간단하게 시각화 한다. 아주 기본적이고 형식적이지 않은 방법으로는 마인드맵, ? 의 방법이 있다. 이러한 방법은 기본적인 구조화 이전, 의뢰자와 같이 온톨로지나 지식그래프, 프로그래밍 자체에 대한 기본적인 지식이 없는 사람들에게 온톨로지에 대해 설명하기에 용이하다.

또한, 본격적인 설계에 앞서 시각화를 통해 단어들간의 전반적인 관계 구조를 파악하고, 이를 기반으로 단어의 재정의나 추가적인 정의 등 본격적인 설계 이전에 단어와 관계를 점검할 수 있다.

2. lightweight ontology ​

경량화 온톨로지란 UML이나 ?? 와 같은, 전 분야의 프로그래밍(주로 객체지향)에서 사용되는 모델링 시각화 방식으로 온톨로지를 시각화하는 방법이다. 대표적으로 UML이 사용된다. UML은 클래스와 인스턴스(객체)를 사각형의 도형으로 표현하고, 도형 안에 클래스의 타입 등의 규칙, 속성 등을 기재한다.

논리적 포함관계는 삼각형으로 채워진 화살표로, 연관관계는 채워지지 않은 화살표로 표현한다.

'포함(substance)'와 '구성(hasComponent)'의 차이 구성요소는 특정 클래스/인스턴스의 속성으로 정의되는, 클래스는 인스턴스를 구성하는 다른 클래스나 인스턴스로, 상속관계와는 관련이 없다. 반면, 포함관계는 특정 클래스에서 파생/상속되는 클래스를 의미한다.

3. OWL ​

OWL은 W3C에서 제정한 표준 온톨로지 모델링 언어이다. UML로 어느정도 구조화된 형식으로 온톨로지 모형을 표현했다면, 이를 온톨로지 모델링 표준 어휘로 더 구체/구조화 된 형식으로 표현한다.

표준화된 어휘란, 즉, 모든 온톨로지 설계자들이 공통적으로 사용하는 어휘라는 것이다. 즉 온톨로지 모델링에 있어 일종의 파이썬이나 자바같은 언어라는 것이다. 아무래도 UML은 온톨로지 모델링 뿐만 아니라 프로그래밍 전반에 사용되는 모델링 언어이다 보니, 온톨로지에서 표준적으로 사용하는 어휘나 관계, 구조나 규칙을 제대로 표현하는데는 한계가 있다.

온톨로지는 OWL로 구조화 됨으로써 기계가 OWL로 작성된 온톨로지 구조를 판독할 수 있게 되고, 모든 온톨로지 설계자들이 어느 도메인의, 어떤 언어 사용자가 설계 했던 간에 이해할 수 있다.

OWL은 기본적으로 RDFS(RDF Schema)의 일종이다. 따라서 RDF 표현 형식을 따라 표현한다.

XML과 마크업 언어

XML은 데이터를 정의하는 규칙을 제공하는 마크업 언어이다. 마크업 언어란 태그 등을 이용해 문서나 데이터 구조를 명기하는 언어로, 프로그래밍 언어에 속하지는 않는다. 컴퓨터 등의 기계에 어떤 계산 작업을 수행하도록 지시하는 언어가 아니라, 단순히 문서/데이터 구조를 표현하는 언어이기 때문이다. XML은 <>를 이용해 누구나 자신만의 문서/데이터 구조를 표현할 수 있도록 한다.

다른 마크업 언어로는 HTML이 있다. HTML은 <h1>등의 태그를 활용해 웹 문서의 구조를 나타낸다.

RDF, RDFS

RDF란 웹 상의 정보를 표현하기 위한 규격이다. HTML이 웹 문서 내용을 구조화한다면, RDF는 웹 문서의 메타 정보를 구조화하여 나타내는 프레임워크이다. RDF는 각기 다른 도메인에서 정의되는 동의어를 의미를 분명하게 구분하기 위해 XML의 namespace를 이용한다. OWL에서는 그래프 구조를 표현하기 위해 RDF 형식을 차용한다.

RDFS는 특정 메타데이터에서 정의하고 있는 어휘들을 선언하기 위해 사용된다. 즉, 어떤 도메인에서 표준적으로 활용하기 위해 도메인에 적합하도록 사용어휘나 표현 규칙을 체계화하여 RDF를 표현형식대로 자료를 기술하는 어휘 체계를 의미한다. RDFS의 일종인 OWL은, 웹 자원에 대한 메타 정보를 온톨로지 형태로 표준화하여 기술하도록 하는 어휘 체계라고 할 수 있다.

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온톨로지의 설계 과정 ​

1. 요구분석과 기초적인 시각화 ​

온톨로지는 요구사항에 맞춰 도메인의 클래스, 인스턴스, 논리적 포함관계(substances), 관계(association)을 정의한다. 이들에 대해 나타내는 용어를 '단어(term)'이라고 하며, 보통 설계 초기에는 단어를 정의하고 이들의 실질적 관계를 엑셀 시트 등의 테이블 형태 문서에 정리한다.

단어에 대한 정의가 완료되면, 이들의 관계를 그래프 구조로 간단하게 시각화 한다. 아주 기본적이고 형식적이지 않은 방법으로는 마인드맵, ? 의 방법이 있다. 이러한 방법은 기본적인 구조화 이전, 의뢰자와 같이 온톨로지나 지식그래프, 프로그래밍 자체에 대한 기본적인 지식이 없는 사람들에게 온톨로지에 대해 설명하기에 용이하다.

또한, 본격적인 설계에 앞서 시각화를 통해 단어들간의 전반적인 관계 구조를 파악하고, 이를 기반으로 단어의 재정의나 추가적인 정의 등 본격적인 설계 이전에 단어와 관계를 점검할 수 있다.

2. lightweight ontology ​

경량화 온톨로지란 UML이나 ?? 와 같은, 전 분야의 프로그래밍(주로 객체지향)에서 사용되는 모델링 시각화 방식으로 온톨로지를 시각화하는 방법이다. 대표적으로 UML이 사용된다. UML은 클래스와 인스턴스(객체)를 사각형의 도형으로 표현하고, 도형 안에 클래스의 타입 등의 규칙, 속성 등을 기재한다.

논리적 포함관계는 삼각형으로 채워진 화살표로, 연관관계는 채워지지 않은 화살표로 표현한다.

'포함(substance)'와 '구성(hasComponent)'의 차이 구성요소는 특정 클래스/인스턴스의 속성으로 정의되는, 클래스는 인스턴스를 구성하는 다른 클래스나 인스턴스로, 상속관계와는 관련이 없다. 반면, 포함관계는 특정 클래스에서 파생/상속되는 클래스를 의미한다.

3. OWL ​

OWL은 W3C에서 제정한 표준 온톨로지 모델링 언어이다. UML로 어느정도 구조화된 형식으로 온톨로지 모형을 표현했다면, 이를 온톨로지 모델링 표준 어휘로 더 구체/구조화 된 형식으로 표현한다.

표준화된 어휘란, 즉, 모든 온톨로지 설계자들이 공통적으로 사용하는 어휘라는 것이다. 즉 온톨로지 모델링에 있어 일종의 파이썬이나 자바같은 언어라는 것이다. 아무래도 UML은 온톨로지 모델링 뿐만 아니라 프로그래밍 전반에 사용되는 모델링 언어이다 보니, 온톨로지에서 표준적으로 사용하는 어휘나 관계, 구조나 규칙을 제대로 표현하는데는 한계가 있다.

온톨로지는 OWL로 구조화 됨으로써 기계가 OWL로 작성된 온톨로지 구조를 판독할 수 있게 되고, 모든 온톨로지 설계자들이 어느 도메인의, 어떤 언어 사용자가 설계 했던 간에 이해할 수 있다.

OWL은 기본적으로 RDFS(RDF Schema)의 일종이다. 따라서 RDF 표현 형식을 따라 표현한다.

XML과 마크업 언어

XML은 데이터를 정의하는 규칙을 제공하는 마크업 언어이다. 마크업 언어란 태그 등을 이용해 문서나 데이터 구조를 명기하는 언어로, 프로그래밍 언어에 속하지는 않는다. 컴퓨터 등의 기계에 어떤 계산 작업을 수행하도록 지시하는 언어가 아니라, 단순히 문서/데이터 구조를 표현하는 언어이기 때문이다. XML은 <>를 이용해 누구나 자신만의 문서/데이터 구조를 표현할 수 있도록 한다.

다른 마크업 언어로는 HTML이 있다. HTML은 <h1>등의 태그를 활용해 웹 문서의 구조를 나타낸다.

RDF, RDFS

RDF란 웹 상의 정보를 표현하기 위한 규격이다. HTML이 웹 문서 내용을 구조화한다면, RDF는 웹 문서의 메타 정보를 구조화하여 나타내는 프레임워크이다. RDF는 각기 다른 도메인에서 정의되는 동의어를 의미를 분명하게 구분하기 위해 XML의 namespace를 이용한다. OWL에서는 그래프 구조를 표현하기 위해 RDF 형식을 차용한다.

RDFS는 특정 메타데이터에서 정의하고 있는 어휘들을 선언하기 위해 사용된다. 즉, 어떤 도메인에서 표준적으로 활용하기 위해 도메인에 적합하도록 사용어휘나 표현 규칙을 체계화하여 RDF를 표현형식대로 자료를 기술하는 어휘 체계를 의미한다. RDFS의 일종인 OWL은, 웹 자원에 대한 메타 정보를 온톨로지 형태로 표준화하여 기술하도록 하는 어휘 체계라고 할 수 있다.

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2. Formalization ​

OWL Properties ​

  1. Annotation Properties

  2. Object Properties

  3. Datatype Properties

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Restriction ​

1. Quantifier Restriction (OWL) ​

2. "hasValue" Restriction (OWL) ​

3 Cardinality Restriction (OWL) ​

4. Existential Restriction ​

5. Universal Restriction ​

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Restriction ​

1. Quantifier Restriction (OWL) ​

2. "hasValue" Restriction (OWL) ​

3 Cardinality Restriction (OWL) ​

4. Existential Restriction ​

5. Universal Restriction ​

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FineTuning 실습코드 ​

깃허브 바로가기

목차 ​

아래 링크를 클릭하면 colab 노트북으로 넘어간다. 혹은 좌측 내비게이션 항목을 클릭해도 된다.

  1. GPT Finetuning
  2. Korquad 데이터셋으로 GPT 파인튜닝 해보기

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Langchain ​

Langchain이란? ​

공식문서 바로가기

  • NLP 관련 기능을 제공하는 파이썬 라이브러리로, 주 목적은 대화형 AI 시스템 구축에 유용한 도구 제공임.

  • 토큰화, 파싱 등의 기본적인 NLP 도구 및, RAG 위한 도구 제공.

  • 따라서, 다른 외부 라이브러리 없이 Langchain을 로드하는 것만으로도 자연어처리부터 RAG, 문서 검색 등 자연어 전반에 대한 task 수행이 가능함.

Lanchain Libraries ​

  • lanchain-core : 기본 추상화 및 랭체인 표현 언어 지원

  • langchain-community : 서드파티 라이브러리 통합

  • langchain : 기본적인 chain, agents, retrieval 지원

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Large Language Model : LLM ​

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🦾 Automatic Differentiation (자동미분) ​

자동미분은 델타논법과 같은 수치 미줌과는 구분되는 미분 방식이다. 수치 미분의 경우 기호위주의 대수학적 규칙을 컴퓨터 연산에 적용하다 보니 비효율적이며, 반올림 에러가 발생한다는 문제가 있다.

따라서, 다수의 input에 대한 미분값을 구해야 하는 컴퓨터 연산 환경에서는 자동미분을 활용한다. 자동미분은 기본적으로 편미분의 연쇄법칙과 같은 연쇄법칙을 적용하여 계산하며, y->x의 순서로 미분 연산을 진행한다.

자동미분은 딥러닝과 머신러닝에서 아주 기본적이고 핵심적인 연산으로, 역전파 방식을 통한 가중치 갱신 과정에 활용된다.

PyTorch와 Tensorflow는 자동미분을 계산하는 라이브러리이며, 이들을 활용해 딥러닝의 신경망을 구현한다. 이 두 라이브러리에서 자동미분은 역전파(backward pass)로 계산된다.

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미적분학 ​

추가자료 ​

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미적분학 ​

추가자료 ​

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1. 미분 ​

미분의 개념 ​

미분은 무수한 점들로 구성된 곡선의 순간 기울기, 즉 곡선의 어느 한 점에서의 기울기를 구하는 것이다. 미분에 대한 표기 방법은 다음과 같다.

1. 일반적 표기 방법 ​

f(x)′

2. 델타 표기 ​

가장 보편적으로 사용되는 표기법이다. 분자에는 미분하는 매개변수를 적어주면 된다.

ddxf(x)

델타논법 (delta method) ​

미분이 "순간적인 기울기"라고 정의했다.즉, 순간이 얼마나 짧은 시간이 되었건 다음 시점의 속도와 현재 시점 속도의 차이값을 구해야 변화율을 구할 수 있는데, 이 찰나의 시간차를 표현한기 위해서 0으로 수렴하는 값을 더해 다음 순간을 정의한다. 여기서 0에 수렴하는 값을 Δx 라고 한다. 즉, 델타논법이란 델타값을 사용해 한 지점의 다음 순간을 정의하고, 다음 순간에서의 값과 현재 값의 차이 값을 두 순간의 차인 델타값으로 나누어 순간 기울기를 구하는 방법이다.

💡극한값

여기서 극한의 개념이 적용되는데, 극한이란 쉽게 말해 어떤 값에 매우 근사하고 있는 값은 사실상 그 값과 다름이 없다는 수학적 약속이다. 그러니까 6에 매우 근사하는 값인 5.999999999를 사실상 6으로 보고 계산하여도 6으로 계산하였을 때의 값과 크게 차이나지 않으니, 이 5.999999를 6으로 두고 계산할 수 있다. 다만, 엄밀히 두 값은 다른 값이니 lim 기호를 써서 6에 근사하는 값임을 나타내야 한다.

델타 논법으로 미분방정식을 표현하면 다음 식과 같다.

f(x)′=limΔx→∞f(x+Δx)−f(x)Δx

미분법칙 ​

1. 상수법칙 ​

상수에 대한 미분값은 0이다.

2. 제곱법칙 ​

제곱의 미분은 제곱되는 값을 상수로 곱하고, 제곱수를 -1 해준다.

3. 상수곱법칙 ​

제곱법칙에서, 제곱되는 값은 변수의 상수곱에 함께 곱해진다.

4. 덧셈법칙 ​

미분시에도 덧셈의 성질은 그대로 유지된다.

5. 곱셈법칙 ​

덧셈과 마찬가지로 곱셈의 성질도 유지된다.

6. 연쇄법칙 ​

가장 중요한 법칙으로, 머신러닝에서 중요한 법칙이다. 변수값으로 함수값이 들어간 복합함수의 경우 (e.g. f(g(x))) 적용되는 법칙이다. 복합함수를 임의의 변수로 치환하여, 해당 변수에 대해 미분한 다음, 변수에 중첩된 함수를 해당 함수의 변수값으로 미분한 값을, 치환값의 미분값에 곱해준다.

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1. 미분 ​

미분의 개념 ​

미분은 무수한 점들로 구성된 곡선의 순간 기울기, 즉 곡선의 어느 한 점에서의 기울기를 구하는 것이다. 미분에 대한 표기 방법은 다음과 같다.

1. 일반적 표기 방법 ​

f(x)′

2. 델타 표기 ​

가장 보편적으로 사용되는 표기법이다. 분자에는 미분하는 매개변수를 적어주면 된다.

ddxf(x)

델타논법 (delta method) ​

미분이 "순간적인 기울기"라고 정의했다.즉, 순간이 얼마나 짧은 시간이 되었건 다음 시점의 속도와 현재 시점 속도의 차이값을 구해야 변화율을 구할 수 있는데, 이 찰나의 시간차를 표현한기 위해서 0으로 수렴하는 값을 더해 다음 순간을 정의한다. 여기서 0에 수렴하는 값을 Δx 라고 한다. 즉, 델타논법이란 델타값을 사용해 한 지점의 다음 순간을 정의하고, 다음 순간에서의 값과 현재 값의 차이 값을 두 순간의 차인 델타값으로 나누어 순간 기울기를 구하는 방법이다.

💡극한값

여기서 극한의 개념이 적용되는데, 극한이란 쉽게 말해 어떤 값에 매우 근사하고 있는 값은 사실상 그 값과 다름이 없다는 수학적 약속이다. 그러니까 6에 매우 근사하는 값인 5.999999999를 사실상 6으로 보고 계산하여도 6으로 계산하였을 때의 값과 크게 차이나지 않으니, 이 5.999999를 6으로 두고 계산할 수 있다. 다만, 엄밀히 두 값은 다른 값이니 lim 기호를 써서 6에 근사하는 값임을 나타내야 한다.

델타 논법으로 미분방정식을 표현하면 다음 식과 같다.

f(x)′=limΔx→∞f(x+Δx)−f(x)Δx

미분법칙 ​

1. 상수법칙 ​

상수에 대한 미분값은 0이다.

2. 제곱법칙 ​

제곱의 미분은 제곱되는 값을 상수로 곱하고, 제곱수를 -1 해준다.

3. 상수곱법칙 ​

제곱법칙에서, 제곱되는 값은 변수의 상수곱에 함께 곱해진다.

4. 덧셈법칙 ​

미분시에도 덧셈의 성질은 그대로 유지된다.

5. 곱셈법칙 ​

덧셈과 마찬가지로 곱셈의 성질도 유지된다.

6. 연쇄법칙 ​

가장 중요한 법칙으로, 머신러닝에서 중요한 법칙이다. 변수값으로 함수값이 들어간 복합함수의 경우 (e.g. f(g(x))) 적용되는 법칙이다. 복합함수를 임의의 변수로 치환하여, 해당 변수에 대해 미분한 다음, 변수에 중첩된 함수를 해당 함수의 변수값으로 미분한 값을, 치환값의 미분값에 곱해준다.

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2. 편도함수 ​

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2. 편도함수 ​

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1. 고유값과 고유벡터 ​

고유벡터와 고유값 ​

고유하다는 것은 상황이 변화해도 그 특성을 잃지 않는 것을 의미한다. 그럼, 벡터가 고유하다는 것은 무엇일까? 벡터가 어떠한 상황에서도 그 특성, 즉 방향성을 잃지 않는 것을 의미한다. 즉, 고유벡터란 선형변환 이후에도 변환 결과가 자신의 상수배를 한 결과일 때의 벡터를 의미한다. 선형 변환이란 쉽게 말해 어떤 행렬을 벡터에 곱하는 것이다.

여러가지 선형변환들

선형변환이란?

벡터에 어떠한 행렬을 곱하는 것을 '벡터에 행렬을 통과시킨다'라고 표현한다. 아무튼, 벡터에 어떠한 행렬을 곱하게 되면 벡터의 크기와 방향이 변한다.

하지만, 아무리 벡터의 크기나 방향이 변해 봤자, 조금 더 벡터의 크기가 커진다거나, 벡터의 방향이 치우치는 정도에 그치게 된다. 벡터가 곡선의 형상을 띄거나 하지는 않는다는 것이다. 이를 '선형적으로 변화했다'라고 말한다.

이처럼, 벡터가 어떠한 행렬을 통과하여 선형적인 변화를 일으키게 하는 것을 선형변환 (linear transformation) 이라고 한다.

선형변환에 속하는 다양한 변환들이 있으며, 모든 변환들은 행렬을 곱하여 이뤄진다. 어찌보면, 벡터에 곱해지는 행렬이 곧 함수와도 같은 역할을 한다고 볼 수 있다. 이런 변환들은 주로 컴퓨터 그래픽에 많이 활용된다. 하지만 아핀변환이라는 것은 LLM에서도 자주 언급되긴 한다.

▶️scailing (비례변환) 비례변환은 벡터의 방향과 크기가 변화하는 변환을 의미한다.

▶️Rotation (회전변환) 회전변환은 좌표평면이 원점을 중심으로 회전하는 것을 의미한다. 회전변환된 벡터는 원래의 벡터와 선형독립이며, 회전변환시 고유값과 고유벡터는 존재하지 않는다.

▶️Shearing (전단변환) 전단변환은 특정 차원값에만 변화를 주는 변환을 의미한다. 기하학적으로 이해하면, y축을 고정하고 x축 방향으로만 변화를 가하는 것을 의미한다.

앞서 살펴본 변환들은 모두 원점이 변화하지 않는 변환이다. 원점을 이동시키는 변환도 있다. 바로 이동변환 이라는 것인데, 대표적으로 아핀변환(Affine) 이 있다.

아핀변환 (Affine Transformation) TBD

식으로 나타내면 다음과 같이 표현할 수 있다.

Xv=λv

여기서 λ 는 벡터에 곱해지는 스칼라를 의미한다. 이 스칼라 값인 람다를 고유값이라고 칭한다. 즉, 고유값이란 고유벡터에 곱해지는 상수값을 의미한다.

고유벡터는 하나만 존재할 수도, 무한하게 존재할 수도 있으며, 고유 벡터가 아예 존재하지 않을 수 있다. 다음은 고유벡터를 구하는 과정을 나타낸 식이다.

Av=λv

Av−λv=0

(A−λI)v=0

v≠0, (A−λI)=0

eigen vector=v∀N(A−λI)

고유벡터가 무한히 존재하는 경우, 보통 해당 벡터의 basis를 고유벡터로 삼는다. 고유벡터가 무한히 존재하는 경우는 보통 행렬식이 0, 즉, 곱해지는 행렬이 선형 종속이라 1차원에서 span하는 경우이다. 이런 케이스를 시각적으로 보면, 좌표 평면이 일직선으로 짜부라드는 것을 확인할 수 있다.

고유값 분해 (eigenvalue decomposition) ​

고유값 분해는 말 그대로 어떤 벡터를 스칼라(고유값)과 고유 벡터로 나누는 것을 의미한다.

λ 값은 여러 개 존재할 수 있으며, 대각행렬로 lambda 값들을 표현할 수 있다. 대각행렬로 나타낸 람다값은 대문자 람다 Λ로 나타낸다.

AV=VΛ 로 나타낼 수 있는데, 고윳값을 갖는 모든 벡터는 Invertable 하다는 성질을 활용해 식을 정리하면

A=VΛV−1 로 정리할 수 있다.

이번에는 Λ만 남도록 식을 정리해보자. 마찬가지로, V의 invertable한 성질을 활용하도록 한다.

V−1A=V−1VΛV−1

V−1AV=V−1ΛV−1V

V−1AV=Λ

다음은 고유값 분해와 관련된 특성들이다.

  1. AT의 고유값은 A의 고유값과 같다.

  2. A가 orthogonal matrix이면, λ=±1 이다.

    A=Q,Qv=λv

    (Qv)TQv=vTQTQv

    QTQ=I,∴(Qv)TQv=||v||22

    이렇게 되면

    λ2v2=||v||22

    ∴λ=±1

    가 성립한다.

  3. A가 Postivie Semi Definite (PSD) 이면 λ는 무조건 0보다 크거나 같다.

Positive Semi Definte란?

  1. Diagonal Matrix Λ의 Non-Zero 고유값의 개수는 rank와 동일하다.

    이 성질은 고유값 분해를 기하학적으로 이해하는데 중요하다.

    rank라는 것은, 벡터의 계수, 즉 벡터가 span하는 차원을 의미한다.

    A는 고유벡터와 고유값의 곱으로 표현된다. 즉, 벡터와 벡터의 스팬하는 비율을 구하는 것이다.

    즉, 고유값 분해란 A의 고유벡터가 span하는 차원들에 대한 벡터로 쪼개고,

    각 차원에 곱해진 λk를 찾는 과정이다.

    여기서 λ는 각 차원별로 곱해지는 값의 모음이므로, 대각행렬성분의 개수가 곧 고유벡터의 랭크와 같다.

    Lambda의 값은 제각각인데, 데이터 압축의 분야에서는 고유값 분해 후, 크기가 작은 고유값은 제거하는 방식으로 데이터를 압축한다.

  2. Symmetric Matrix는 무조건 Diagonalizable 하며 (역 성립 X), 따라서 A=QΛQT 된다.

    대칭행렬이란 AT=A인 행렬이다. 대칭행렬은 무조건 대각화가 가능하다는 성질을 갖는다.

(+) 대각화란?(diangolize)

대각화란 어떠한 행렬을 고유벡터로 이뤄진 가역행렬과 가역행렬에 곱해진 고유값들에 대한 대각행렬의 곱으로 나타내는 것을 의미한다. 즉, AX=XΛ,A=XΛXT의 꼴로 나타내는 것을 의미한다.

대각화가능조건

  1. n x n 행렬 A는 일차독립인 교유벡터를 가져야 한다. 즉, 행렬 A의 고유벡터들은 Full-rank여야 한다.
  2. n x n 행렬 A가 서로 다른 n개의 고유값을 갖는 경우 대각화 가능하다. 고유값분해와 혼동하지 말아야 할 것은, 고유값분해는 꼭 서로 다른 고유값을 가질 필요는 없다는 것이다.(Λ의 대각성분으로 0이 대다수 나타나는 경우가 있다.)

고유값 분해의 장점 ​

고유값 분해로 얻을 수 있는 장점은 무엇일까?

  1. 행렬의 거듭제곱 계산이 용이해진다

    고유값분해가 되는 행렬을 거듭제곱하면 다음과 같이 나타낼 수 있다.

    Ak=VΛV−1⋅VΛV−1⋅VΛV−1...

    여기서 V−1V는 I로, 나열된 수식에서 소거된다. 따라서, 이를 정리하면

    Ak=VΛkV−1

    을 얻을 수 있는데, Λk는 대각행렬의 제곱이므로 복잡한 연산 없이 대각성분들을 k승 해주기만 하면 된다.

  2. 고유값 분해를 통해 쉽게 Inverse Matrix를 얻을 수 있다

    A−1=(VΛV−1)−1

    VΛ−1V−1

  3. 행렬식을 구하기 쉽다

    A의 행렬식을 고유값과 고유벡터의 곱으로 나타내면

    det(A)=det(VΛV−1)

    det(A)=det(V)det(Λ)det(V−1)

    인데, invese 벡터의 행렬식은 벡터의 행렬식의 약수이다.

    따라서, 결국에는 det(A)=det(Λ) 만 남는다.

    그리고, 대각행렬에서의 행렬식은 곧 대각성분들의 총 곱과 같으므로

    det(A)=Πi=1nλi

    가 성립한다.

  4. trace(대각합)값을 구하기 쉽다

    대각합은 대각선상의 성분들을 더하는 것이다.

    tr(A)=tr(VΛV−1)이고, 대각합의 성질 상, tr(VΛV−1)=tr(ΛVV−1) 로도 정리할 수 있다.

    tr(ΛVV−1)=tr(Λ) 이므로, tr(A)=tr(Λ)이다.

    따라서,

    tr(A)=tr(Λ)=Σi=1nλi

    가 성립한다. 즉, A의 대각합은 고유값의 합과 동일하다는 것이다.

  5. rank-deficient인 행렬일 경우, 0인 고유값이 하나 이상 존재함을 알 수 있다.

    rank-deficient는 선형 종속인 행렬임을 파악할 수 있는 요소이다. rank-deficient인 경우, $$\lambda$$는 0인 고유값을 반드시 갖게 되는 성질이 있다. 직관적으로 생각하면, full rank가 아닌 이상, 백터의 스팬 차원이 하나 줄어든다.

    따라서, 고유벡터를 각 차원 공간에서 스팬하도록 하는 행렬 $$\Lambda$$도 원본 행렬과 동일한 크기의 대각행렬이 되겠지만, 스팬하지 않는 차원의 대각 성분값은 0이 될 수 밖에 없는 것이다.

    이를 수식으로 증명하면 다음과 같다.

    A=[1236],v=[xy]Av=λv[1236]∗[xy]=λ∗[xy][x+2y3x+6y]=λ∗[xy]=[λxλy]

    2y=(λ−1)x

    로 정리할 수 있다. 이를 $$3x+6y = \lambda y$$ 에 대입하면,

    3x+3(λ−1)x=λ((λ−1)x/2)

    6λ=(λ2−λ)x

    6=(λ−1)x

    x=6/(λ−1)

    이를 다시 원래 식에 대입하면

    2y=(λ−1)x

    (λ−1)∗6/(λ−1)=6

    ∴y=3

    를 구할 수 있다.

    이를 다시 식에 대입하면 첫 번째 방정식에선 다음의 값이 도출된다.

    x+6=λx

    (λ−1)x=6

    두 번째 방정식에선 다음의 식이 도출된다.

    3x+6∗3=λ∗3

    x+6=λ,x=λ−6

    두 번째 방정식에서 얻은 x의 값을 첫 번째 방정식에 대입하면

    3x=3λ−18

    x=λ−6

    x를 앞서 정리한 2y=(λ−1)x,y=3에 대입하면

    (λ−1)(λ−6)=6

    λ−7λ+6=6

    λ(λ−7)=0

    위의 식에서 λ는 0 또는 7의 값을 갖는 것을 확인 할 수 있다.

rank-deficient란? rank는 행렬이 갖는 독립적인 행이나 열의 개수를 의미한다. 정방행렬에서 full rank는 랭크가 n개, n x m 행렬에서 full-row rank는 rank가 n, full-column rank에서 rank는 m이다. rank-deficient는 n x m 행렬에서 rank 값이 min(n,m) 보다 작은 경우를 의미한다. 즉, 행렬의 벡터가 독립적이지 않아, 행렬의 차원 수 만큼 벡터가 표현될 수 없는 상태를 의미한다. rank-deficient는 따라서 선형 종속인 상태를 의미하기도 하므로, rank-deficient인 행렬의 행렬식은 0이 될 수밖에 없다.

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2. 특이값과 특이값 분해 ​

intro ​

고유값분해는 실수인 고유값과 직교벡터인 고유벡터를 가지도록 정사각행렬을 분해한다. 고유값분해를 했을 때, 모든 행렬이 완벽하게 실수 고유값과 직교인 고유벡터를 갖는 것은 아니다. 즉, 고유벡터는 모든 형태의 행렬에 적용할 수 없다는 한계점을 갖는다.

반면, 특이값 분해는 모든 형태의 행렬에 적용할 수 있다는 장점이 있다.

특이값, 좌특이벡터, 우특이벡터 | 선형대수에서 "특이(singular)"가 갖는 의미는 행렬식이 0인 정사각 행렬을 의미한다. 즉, 정방 행렬이나 역행렬이 존재하지 않는 행렬을 의미한다. 그리고 앞으로 다룰 특이값 분해에서 등장하는 좌특이벡터, 우특이벡터는 각각 원래 행렬 A의 열, 행을 선형독립인 기저벡터로 변환해주는 역할을 한다. 여기서 좌특이벡터는 U 행렬의 열벡터들을 의미하고, 우특이벡터는 V 행렬의 행벡터를 의미한다. 좌특이벡터와 우특이벡터는 직교하는 성질을 갖고 있다.

데이터 사이언스에서 특이값 분해는 다방면에서 활용된다. 그 이유는 특이값분해는 어떠한 행렬이던 간에 행렬을 랭크 1인 조각으로 나눌 수 있으며, 나뉜 조각들이 중요한 순서대로 나온다는 특성을 갖기 때문이다.

조각 σ1u1v1T가 A에 가장 가까운 랭크 1 행렬이라고 할 수 있다.

특이값분해란? ​

특이값분해는 행렬을 좌특이벡터(left singular vector)와 우특이벡터(right singular vector), 고유값으로 분해하는 것이다. 좌특이벡터와 우특이벡터는 고유값 σ에 대응하는 기저라고 이해할 수 있다. 수식으로 보면 A=UΣVT 이다. $$U$$는 좌특이벡터로 구성된 행렬이고, V는 우특이벡터로 구성된 행렬이다. 각 벡터 uk,vk는 직교하는 성질을 갖는다. (이 둘이 기저벡터임을 생각하면 자명하다.)

고유값 분해와 유사한 지점이 많으나, 특이값 분해는 ATA 와 AAT가 무조건 대칭행렬이라는 점을 활용해, 행렬이 선형변환을 거쳐도 방향성을 잃지 않는 고유적인 특성 벡터를 구한다. 다만, 행렬이 정방행렬이 아니므로, 두 개의 특이벡터를 구하게 된다.

ATA와 AAT가 대칭행렬임을 확인하는 식은 다음과 같다.

(ATA)T=ATA(AAT)T=AAT

ATA 나 AAT 모두 전치해도 자기 자신과 같으므로 대칭적이라는 것이다.

특이값 분해 공식 ​

기본적으로 특이값 분해는 $$A = U \Sigma V^T$$ 를 구하는 것이다. 이때, A에 A의 전치 행렬을 곱해 대각화 가능한 정방행렬로 만들어준다. 대각화가 가능하다는 것은 곧 분해가 가능한 형태라는 것이다.

특이값분해

ATA 와 AAT는 다른 행렬이나, 공통적으로 둘 다 대칭적이다. 특이값 분해는 이 점을 활용해, 두 식으로부터 도출되는 특이벡터와 특이 벡터가 통과하는 시그마 행렬을 찾아낸다.

우선 ATA로 특이값 분해하는 과정을 알아보자. 그 전에 특이값 분해가 가능하도록 하는 조건 3개를 알아보자.

특이값 분해의 조건

  1. V는 A의 정규직교 고유벡터를 포함한다.
  2. U는 A의 정규직교 고유벡터를 포함한다.
  3. σ1.σ2...σk는 ATA,AAT 모두의 0이 아닌 고유값이다

특이값 분해는 A를 ATA,AAT의 대칭행렬을 좌특이벡터, 우특이벡터와 두 대칭행렬에서 공통적으로 갖는 고유값 행렬로 분해한다.

AAT=UΣVT⋅VΣTUT

VT⊥V,∴V⋅VT=I

∴AAT=UΣΣTUT

U는 전치벡터와 직교하며 ΣΣT는 U 의 크기를 갖는 정방행렬이 된다.

따라서, 사실상 QΛQT의 형태를 띄게 된다. 여기서부턴 고유분해하듯 구하면 된다.

다음으로는 ATA로 특이값 분해하는 과정을 알아보자.

ATA=VΣTUT⋅UΣVT

UT⋅U=I,∴ATA=VTΣTΣV

AAT의 식과 유사한 형태의 값이 나온다. 여기서도 V는 U와 마찬가지로 직교벡터이고, 시그마 행렬은 V의 크기를 따르는 정방행렬이다. V도 U와 동일한 방식으로 구해준다.

  • ΣΣT는 U의 차원수와 동일한 크기를 갖는 고유값의 정방행렬을 가진다.
  • 반대로 ΣTΣ는 VT와 동일한 크기를 갖는 고유값의 정방행렬을 갖는다.

Σ는 VT와 U가 공통적으로 통과하는 행렬로, ΣTΣ,ΣΣT의 대각성분은 σn의 제곱으로 이뤄져 있다. dimension(U)>dimension(VT) 일 때, ΣΣT는 n개의 0이 아닌 U, V 모두의 고유값으로 이뤄진 대각성분을 가지며, m-n개의 나머지 대각성분은 0의 값을 갖는다.

무어-펜로즈 유사역행렬(Moore-Penrose Pseudo Matrix Inverse) ​

무어-펜로즈 유사역행렬(:의사역행렬)은 임의 행렬에 A에 대해서 , n > m (데이터개수 > 파라미터)이고 모든 열벡터가 선형 독립일 때 다음의 식이 성립한다. n >m 인 상태는 과결정(overdermined)상태를 의미하기도 한다. 선형회귀분석이 적용되는 아주 일반적인 케이스이다

A+=(ATA)−1AT

A+A=(ATA)−1ATA

A+A=(ATA)−1(ATA)=I

∴A+A=I

이때, ATA는 가역행렬이다.

A+가 좌측 역행렬이 되는 것을 의미한다.

반대로 n < m의 경우 A+는 우측역행렬이 된다. n < m(파라미터 > 데이터)은 불충분결정평면 (underdetermined) 상태를 의미한다. 회귀에서 자주 보이는 유형은 아니나, 딥러닝에서는 이러한 형태의 행렬벡터가 빈번히 등장한다. 딥러닝의 feature(파라미터)는 수억수천만개이지만 학습시킨 데이터는 이 수준에는 못 미치기 때문이다.

A+=AT(AAT)−1

AA+=AAT(AAT)−1

AA+=(AAT)(AAT)−1=I

∴AA+=I

A의 의사역행렬을 구한는 방식은 특이값 분해를 거친다.

A=UΣVT로 분해되었을 때, A+ 구해진 U, V를 바탕으로 다음의 공식을 통해 구해진다.

A+=VΣ+UT

이때, Σ+는 Σ는 특이값을 대각선상에 표현한 대각행렬이므로 유사역행렬은 단순히 특잇값들에 역수를 취하는 방식으로 구할 수 있다. 0인 값은 그냥 0으로 둔다.

선형회귀에서의 의사역행렬 활용 ​

선형회귀라는 것은 기본적으로 독립변수(x)에서 종속변수(y)를예측하기위한방법으로독립변수와가중치벡터(w$)의 가중합으로 y와 가장 근사한 값 y^을 계산하는 것을 의미한다.

선형회귀모형을 수식적으로 재현하면 다음과 같다.

y^=wTx

예측의 정확성은 가중치벡터가 관건이라고 할 수 있다. 통상적인 관점에서의 머신러닝은 대체로 이 가중치의 최적값을 최대한 효율적으로 구하는 것이 목표이다.

가중치는 잔차제곱합(Residual Sum of Square : RSS)으로 구할 수 있다. 여기서 잔차라는 것은 예측값과 실제값의 차이, 즉 오차(error : e)를 의미한다. 목표는 간단하다. 잔차들의 합이 최소가 되도록 하는 것이다.

Σi=1Ne2=Σi=1N(yi−wTxi)2

이 식을 좀 더 대수적관점에 접근해보면

e2=eTe

eTe=(y−Xw)T(y−Xw)

로 볼 수 있다.

대체로 머신러닝에서는 방정식의 수보다 미지수가 훨씬 많기에, 최소자승법을 통해 해결한다. 즉, 잔차벡터의 크기를 최소화하는 가중치벡터를 찾는 것이다.

잔차의 제곱을 취한 벡터인 Σe2는 L2 Norm으로도 볼 수 있다. 즉, 최소자승법을 통한 해법은 곧 잔차 벡터의 norm을 최소화 하는 문제이다.

최소자승법에서 Ax≈b이다. 이러한 관점에서 $$x$$를 가중치와 정답값의 결합으로 표현하는 과정은 다음과 같다.

Ax=b

ATAx=ATb

(ATA)−1ATAx=(ATA)−1ATb

P(ATA)−1(ATA)x=(ATA)−1ATb

x=(ATA)−1ATb

뭔가 익숙한 식이 보이지 않는가? 우항의 (ATA)−1AT를 보자. 앞서 유사역행렬에 대한 설명에서 A의 유사역행렬이 (ATA)−1AT임을 공부했다.

따라서, 이 식은

x=A+b

로 나타낼 수 있다. 이렇게 하면 보다 간단하게 선형회귀 방정식을 구할 수 있다. 최적의 잔차를 구할 때까지 가중치 벡터를 갱신할 필요가 없어진다는 것이다.

Ref ​

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3. PCA (TBD) ​

PCA란 라벨링 되지 않은, 고차원의 데이터 집합이 주어졌을 때, 데이터의 특성을 최대한 유지하여 낮은 차원으로 데이터를 압축하는 것을 의미한다.

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- + \ No newline at end of file diff --git a/docs/.vitepress/dist/contents/MATH/linear-algebra-application/intermediate-chap-4.html b/docs/.vitepress/dist/contents/MATH/linear-algebra-application/intermediate-chap-4.html index 597307b..1d293e6 100644 --- a/docs/.vitepress/dist/contents/MATH/linear-algebra-application/intermediate-chap-4.html +++ b/docs/.vitepress/dist/contents/MATH/linear-algebra-application/intermediate-chap-4.html @@ -63,7 +63,7 @@
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4. 선형회귀와 다중회귀 ​

선형회귀 ​

선형회귀란 다수의 데이터, 즉 독립변수 x로부터 종속변수(레이블값) y를 예측하기 위한 통계적 기법이다. 선형회귀란, 독립변수에 따른 종속변수의 분포에 따라, 독립변수로부터 종속변수가 도출될 수 있는 선형 함수 wx+b를 찾아 내는 것이 목표이다.

w,b를 찾아내는 것이 관건인데, 선형회귀 모델에서는 잔차제곱합(Residual Sum of Square)를 통해 w,b의 예측값에 따라 도출된 예측값인 y^와 실제 값 y의 차이인 오차값 e가 최소화 되는 값을 찾음으로서 두 값을 구한다.

모든 독립변수 x에 대한 에러값 총합에 대한 식은 다음과 같다.

Σ1Ne2=Σ1N(yi−(wxi+b))2

에러가 최저가 되는 지점은 각각 w, b에 대해 에러 식을 미분하면 구할 수 있다.

  • w에 대해 미분

σσwΣ1Ne2=σσwΣ1N(yi−(wxi+b))2

σσwΣ1Ne2=Σ1N−2xi(yi−(wxi+b))=0

Σ1N(2xi2)w−2xi(yibi+1)b=0

Σ(2xi2)w−2Σxi(yib+1)b=0

  • b에 대해 미분

σσbΣ1Ne2=Σ1N(yi−(wxi+b))=0

Σ1N(−xi)w+(yib−1)b=0

Σ(−xi)w+Σ(yib−1)b=0

합에 대한 식의 연립은 역행렬 연산을 통해 구할 수 있다.

[Σ(2xi2)Σ(xi(yibi+1))Σ(−xi)Σ(yib−1)][wb]=[00]

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- + \ No newline at end of file diff --git a/docs/.vitepress/dist/contents/MATH/linear-algebra-application/intermediate-chap-5.html b/docs/.vitepress/dist/contents/MATH/linear-algebra-application/intermediate-chap-5.html index 7250554..899307e 100644 --- a/docs/.vitepress/dist/contents/MATH/linear-algebra-application/intermediate-chap-5.html +++ b/docs/.vitepress/dist/contents/MATH/linear-algebra-application/intermediate-chap-5.html @@ -12,7 +12,7 @@ - + @@ -47,7 +47,7 @@ - + @@ -62,8 +62,8 @@ -
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5. 로지스틱 회귀 ​

'XX 회귀' 라고 하는 것은, 데이터를 통해 'XX'를 구하는 것이라고 이해할 수 있다. '선형 회귀'에서는 말 그대로 '선'(정확히는 선형방정식)을 찾았다. 로지스틱 회귀에서는 'logistic(기호논리)'을 찾는 것이 목표이다. 그러니까, 어떤 기호값(label)을 분류하는 논리식을 찾는 것이 로지스틱 회귀이다. 논리식이래서 온톨로직한 문장을 찾는건 아니고, 결국에는 선형회귀와 마찬가지로 입력 데이터세트 X를 결정 공간에 잘 사상시킬 수 있는 사상 벡터 W 를 찾는게 목표이다.

예를들어 키, 몸무게에 따른 남/여 카테고리 컬럼이 존재하는 경우, 남성과 여성을 구분하는 결정 직선을 구하고자 한다면 로지스틱 회귀를 적용해야 한다.

최소제곱오차에 따른 선형 회귀 방식을 적용하는 경우는 키에 따른 몸무게를 예측하고자 하는 상황과 같이, 이미 존재하는 두 수치 변수간의 관계를 파악하고자 할 때 이다.

위의 사례들에서 알 수 있다시피, 특정 입력 값에 대한 출력값을 예측하는 모델(function)을 구하고자 할 때는 선형회귀를, 레이블이 존재 할 때 입력값들을 구분하고자 하는 기준 function을 구하고자 할 때는 로지스틱 회귀를 적용한다.

그런데 이제, 선형회귀에서는 결정 공간(치역)이 연속적인 값들로 구성되어 있지만, 로지스틱에서는 결정 공간이 레이블 값(흔히 0, 1)로만 구성되어 있다. 그래서 로지스틱 회귀에서는 시그모이드 함수에 사상된 벡터를 넣어서 0~1사이의 값으로 바꿔버린다.

엄밀히 따지면, 로지스틱 회귀에서 wx+b, 즉 WT⋅X 갖는 의미는 W와 X의 내적을 의미하고, 내적이라는 것은 곧 W가 X의 성질을 얼마나 잘 반영하고 있는지(닮았는지)를 의미한다. 즉, W라는 것은, 어떤 특정한 레이블을 갖는 데이터 집합에 대한 벡터라고 할 수 있다.

따라서 wtx가 직교, 즉 0의 값을 가지면 완전히 상이함을, 양/음의 값을 가지면 x 벡터와 w벡터가 어느정도 유사성을 띄고 있음을 의미한다. 여기에 시그모이드 함수를 적용함으로써 계산된 값을 확률적으로 표현하게 된다.

💡시그모이드 함수와 우도

시그모이드 함수 ​

시그모이드 함수 역시 입력 값을 특정 값으로 사상시키는 함수로, 전체 실수 값을 0~1 사이의 값으로 변환해주는 함수이다.

Sigmoid=11+exp(−z), z=wx+b

w는 시그모이드 곡선의 기울기를, b는 시그모이드 값의 중앙값에 대한 x값을 결정한다.

우도 ​

시그모이드 함수는 입력값에 대해 0~1 사이의 값을 변환한다. 0~1사이의 값은 곧 확률값의 형태로 볼 수 있으나, 어쨌든 변환된 모든 값의 총합이 1이 되리라 보장하지 않으므로 확률밀도함수 모델을 적용해 구할 수는 없고, 대신 이들의 곱을 통해 '그럴싸한 정도'인 우도를 구할 수 있다.

그렇다면 최적의 function은 곧 우도값을 최대로 만드는 함수일 것이다. 쉽게 생각하면, 시그모이드 함수에 로그를 취한 후 미분을 하는 방법이 있겠다. 하지만 이렇게 되면 실제 레이블값이 어찌 되었건 w벡터는 단순히 데이터세트 벡터 세트 X와의 유사성을 최소로 하는 벡터가 될 것이다.

이것이 문제가 되는 경우는 특정 레이블 그룹에서 극단적인 x가 존재하는 상황이다. 온전히 데이터값의 분포로만 W를 구하면 분류가 부정확할 것이다.

따라서, 로지스틱 회귀에서는 우도값이 최대, 즉 '그럴싸 함'이 최대가 되는 W를 구한다. 최대우도 L(w)를 구하는 식은 아래와 같다. 확률의 곱으로 최대 우도를 구한다.

L(w)=p(t(1),...,t(N)|x(1),...,x(N);w)=∏i=1Np(t(i)|x(i);w)p(t(i)|x(i);w)=(1−p(C=1|x(i);w))t(i)p(C=0|x(i);w)1−t(i)p(C=0|x(i);w)=11+exp(−z), z=wx+b

위 식은 학습 데이터가 iid 조건을 충족한다는 전제에서 성립한다. 이렇게 되면, 옳은 레이블이 아니나 시그모이드값이 크게 나오는 경우라도 지수값이 0이 되면서 해당 시그모이드 값이 무시되고 반대의 확률값만 곱해지게 된다.

최대의 L(w)를 구하기 위해선 미분을 적용하기 위해 자연로그를 취한다. 이렇게 하면 w에 대해 미분이 가능해진다. 여기서 더 나아가, 계산의 편리함을 위해 로그 값에 음수를 곱한다. 그러면 기울기가 최소가 되는 지점을 구하게 된다.

여기서 목적함수가 되는 비용함수(cost function)는 −logL(w)이다. 비용함수가 최소가 되는 w값이 곧 로그 우도를 최대로 하는 함수이다.

💡왜 음수로 바꾸는가

사실 로그 값에 대해 최대가 되는 w를 찾는 것도 동일한 결과를 도출한다. 다만 굳이 음수값으로 바꿔 구하는 이유는 목적함수를 비용함수로 정의했기 때문이다. 비용함수의 goal이 최소지점이므로, 이러한 관점에 맞춰 값을 구하기 위해 굳이 음수로 값을 취해 구하는 것이다. 구태여 비용함수로 정의해 구하는 이유는 최저값 계산이 더 간편하기 때문이라고 한다.

따라서, L(w)에 대한 최종적인 수식은 아래와 같다.

L(w)=∏i=1N(1−11+e−z(i))t(i)(11+e−z(i))1−t(i)L(w)=∏i=1N(e−z(i)1+e−z(i))t(i)(11+e−z(i))1−t(i)=∏i=1N(e−z(i)1+e−z(i))t(i)(11+e−z(i))−t(i)(11+e−z(i))1=∏i=1N(e−z(i))t(i)(11+e−z(i))1=∏i=1N(e−z(i))t(i)(1+e−z(i))−1

로그를 씌워 최종적으로 정리하면 아래와 같다.

logL(w)=l(w)=−∑i=1N(t(i)z(i))−∑i=1Nloge(1+e−z(i))

비용함수의 형태로 정의하면 다음과 같다. 이제부터는 이 비용함수를 w에 대해 미분하도록 하겠다.

−l(w)=∑i=1N(t(i)z(i))+∑i=1Nloge(1+e−z(i))

이제 wi에 대한 편미분을 통해 최적의 W 벡터를 찾으면 된다. 그런데 여기서 문제가 있다. w에 대해 미분해야 하는데, 식은 치환변수인 z로 작성되어 있다. 따라서 여기서는 연쇄법칙에 따른 매개변수 미분을 진행한다.

이때, xi의 차원수만큼 wi가 존재할 것이고, wi 하나를 찾기위해선 N개의 데이터에 대해 연산해야 한다. 즉, w1...wD의 미분계수가 존재하고, 이는 미분계수 벡터의 형태로 나타낼 수 있다.

▽W=[δδw1,δδw1,...,δδwD]T

이것을 이제 데이터 세트 하나씩 곱하여 미분하면 된다. 아마 벡터 연산으로 나타내면 아래와 같은 모양일 것이다. 모든 식에서 0이 나오도록 연립방정식을 풀면 된다.

우행렬의 열벡터 각각은 i번째의 x 벡터이다. 보통은 벡터연산으로 z=WTXi로 풀기에, 아래와 같이 행렬로 연산한다.

[δδw1δδw2...δδwD][t(1)w1x1(1)+log(1+e−w1x1(1))...t(N)w1x1(N)+log(1+e−w1x1(N))t(1)w2x2(1)+log(1+e−w2x2(1))...t(N)w2x2(N)+log(1+e−w2x2(N))...t(1)wDxD(1)+log(1+e−wDxD(1))...t(N)wDxD(N)+log(1+e−wDxD(N))]=▽WT⋅[t(1)WTX1+log(1+e−WTX1)...t(N)WTXN+log(1+e−WTXN)]

여기서 δδw1[t(1)w1x1(1)+log(1+e−w1x1(1))]의 식만 계산 해보자.

💡로그의 미분

case1 y=ln x

dydx=1x

case2 y=loga x

dydx=1x ln a

case3 y=ln kx

dydx=1x

이제부터는 특정 스칼라 값들에 대한 연산이므로 w1=w,x1(1)=x로 간단하게 표현하겠다.

자연로그가 있으므로, 자연로그 내부의 식은 u로 치환하여, 연쇄 법칙을 적용하여 미분해야 한다.

1. w에 대해 미분하기

σσwtwx+σσwln(u), u=1+e−wxtx+σσuln(u)σuσw

2. u에 대해 미분하기

σσuln(u)=1u

3. u를 w에 대해 미분하기

σuσw=−xe−wx

4. 대입하기

tx+(1+e−wx)−1(−xe−wx)=x(t−(1+e−wx)−1(e−wx))

5. 미분값이 0이 되는 지점의 w 구하기

  • t=1 인 경우
x+(1+e−wx)−1(−xe−wx)=x(1−(1+e−wx)−1(e−wx))=01−e−wx(1+e−wx)=0e−wx(1+e−wx)=1e−wx=1+e−wx−wx=ln(1+e−wx)w=−ln(1+e−wx)x

exponential의 미분은 매우 까다롭다. 그런데, 이런 exponential에 대한 미분을 NxD회 실행해야 하므로, 하나의 CPU를 이용해 순차적으로 미분방정식을 푸는 것은 매우 비효율적이고, 사실상 불가능한 일일 것이다.

따라서, 딥러닝의 학습 시에는 그래픽 연산에 최적화된 프로세서인 GPU를 이용해, dn에 대해 동시다발적으로 다수의 데이터에서 미분이 계산될 수 있도록 병렬연산을 진행한다. 즉, t1의 시간에 N개의 w1에 대한 미분 연산이 동시에 진행된다는 것이다. GPU 개수가 많으면 많을 수록, 동시에 계산할 수 있는 차원 수도 많아지므로 훨씬 빨라진다.

물론, 제아무리 GPU와 같은 하드웨어의 도움을 받아 연산을 효율화 할 수 있다고 하더라도, 기본적으로는 차원 수를 줄인다던가, 데이터 샘플링을 통해 대표성을 갖는 데이터세트에 대해서만 학습을 진행한다던가 하는 방법을 함께 사용한다.

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5. 로지스틱 회귀 ​

'XX 회귀' 라고 하는 것은, 데이터를 통해 'XX'를 구하는 것이라고 이해할 수 있다. '선형 회귀'에서는 말 그대로 '선'(정확히는 선형방정식)을 찾았다. 로지스틱 회귀에서는 'logistic(기호논리)'을 찾는 것이 목표이다. 그러니까, 어떤 기호값(label)을 분류하는 논리식을 찾는 것이 로지스틱 회귀이다. 논리식이래서 온톨로직한 문장을 찾는건 아니고, 결국에는 선형회귀와 마찬가지로 입력 데이터세트 X를 결정 공간에 잘 사상시킬 수 있는 사상 벡터 W 를 찾는게 목표이다.

예를들어 키, 몸무게에 따른 남/여 카테고리 컬럼이 존재하는 경우, 남성과 여성을 구분하는 결정 직선을 구하고자 한다면 로지스틱 회귀를 적용해야 한다.

최소제곱오차에 따른 선형 회귀 방식을 적용하는 경우는 키에 따른 몸무게를 예측하고자 하는 상황과 같이, 이미 존재하는 두 수치 변수간의 관계를 파악하고자 할 때 이다.

위의 사례들에서 알 수 있다시피, 특정 입력 값에 대한 출력값을 예측하는 모델(function)을 구하고자 할 때는 선형회귀를, 레이블이 존재 할 때 입력값들을 구분하고자 하는 기준 function을 구하고자 할 때는 로지스틱 회귀를 적용한다.

그런데 이제, 선형회귀에서는 결정 공간(치역)이 연속적인 값들로 구성되어 있지만, 로지스틱에서는 결정 공간이 레이블 값(흔히 0, 1)로만 구성되어 있다. 그래서 로지스틱 회귀에서는 시그모이드 함수에 사상된 벡터를 넣어서 0~1사이의 값으로 바꿔버린다.

엄밀히 따지면, 로지스틱 회귀에서 wx+b, 즉 WT⋅X 갖는 의미는 W와 X의 내적을 의미하고, 내적이라는 것은 곧 W가 X의 성질을 얼마나 잘 반영하고 있는지(닮았는지)를 의미한다. 즉, W라는 것은, 어떤 특정한 레이블을 갖는 데이터 집합에 대한 벡터라고 할 수 있다.

따라서 wtx가 직교, 즉 0의 값을 가지면 완전히 상이함을, 양/음의 값을 가지면 x 벡터와 w벡터가 어느정도 유사성을 띄고 있음을 의미한다. 여기에 시그모이드 함수를 적용함으로써 계산된 값을 확률적으로 표현하게 된다.

💡시그모이드 함수와 우도

시그모이드 함수 ​

시그모이드 함수 역시 입력 값을 특정 값으로 사상시키는 함수로, 전체 실수 값을 0~1 사이의 값으로 변환해주는 함수이다.

Sigmoid=11+exp(−z), z=wx+b

w는 시그모이드 곡선의 기울기를, b는 시그모이드 값의 중앙값에 대한 x값을 결정한다.

우도 ​

시그모이드 함수는 입력값에 대해 0~1 사이의 값을 변환한다. 0~1사이의 값은 곧 확률값의 형태로 볼 수 있으나, 어쨌든 변환된 모든 값의 총합이 1이 되리라 보장하지 않으므로 확률밀도함수 모델을 적용해 구할 수는 없고, 대신 이들의 곱을 통해 '그럴싸한 정도'인 우도를 구할 수 있다.

그렇다면 최적의 function은 곧 우도값을 최대로 만드는 함수일 것이다. 쉽게 생각하면, 시그모이드 함수에 로그를 취한 후 미분을 하는 방법이 있겠다. 하지만 이렇게 되면 실제 레이블값이 어찌 되었건 w벡터는 단순히 데이터세트 벡터 세트 X와의 유사성을 최소로 하는 벡터가 될 것이다.

이것이 문제가 되는 경우는 특정 레이블 그룹에서 극단적인 x가 존재하는 상황이다. 온전히 데이터값의 분포로만 W를 구하면 분류가 부정확할 것이다.

따라서, 로지스틱 회귀에서는 우도값이 최대, 즉 '그럴싸 함'이 최대가 되는 W를 구한다. 최대우도 L(w)를 구하는 식은 아래와 같다. 확률의 곱으로 최대 우도를 구한다.

L(w)=p(t(1),...,t(N)|x(1),...,x(N);w)=∏i=1Np(t(i)|x(i);w)p(t(i)|x(i);w)=(1−p(C=1|x(i);w))t(i)p(C=0|x(i);w)1−t(i)p(C=0|x(i);w)=11+exp(−z), z=wx+b

위 식은 학습 데이터가 iid 조건을 충족한다는 전제에서 성립한다. 이렇게 되면, 옳은 레이블이 아니나 시그모이드값이 크게 나오는 경우라도 지수값이 0이 되면서 해당 시그모이드 값이 무시되고 반대의 확률값만 곱해지게 된다.

최대의 L(w)를 구하기 위해선 미분을 적용하기 위해 자연로그를 취한다. 이렇게 하면 w에 대해 미분이 가능해진다. 여기서 더 나아가, 계산의 편리함을 위해 로그 값에 음수를 곱한다. 그러면 기울기가 최소가 되는 지점을 구하게 된다.

여기서 목적함수가 되는 비용함수(cost function)는 −logL(w)이다. 비용함수가 최소가 되는 w값이 곧 로그 우도를 최대로 하는 함수이다.

💡왜 음수로 바꾸는가

사실 로그 값에 대해 최대가 되는 w를 찾는 것도 동일한 결과를 도출한다. 다만 굳이 음수값으로 바꿔 구하는 이유는 목적함수를 비용함수로 정의했기 때문이다. 비용함수의 goal이 최소지점이므로, 이러한 관점에 맞춰 값을 구하기 위해 굳이 음수로 값을 취해 구하는 것이다. 구태여 비용함수로 정의해 구하는 이유는 최저값 계산이 더 간편하기 때문이라고 한다.

따라서, L(w)에 대한 최종적인 수식은 아래와 같다.

L(w)=∏i=1N(1−11+e−z(i))t(i)(11+e−z(i))1−t(i)L(w)=∏i=1N(e−z(i)1+e−z(i))t(i)(11+e−z(i))1−t(i)=∏i=1N(e−z(i)1+e−z(i))t(i)(11+e−z(i))−t(i)(11+e−z(i))1=∏i=1N(e−z(i))t(i)(11+e−z(i))1=∏i=1N(e−z(i))t(i)(1+e−z(i))−1

로그를 씌워 최종적으로 정리하면 아래와 같다.

logL(w)=l(w)=−∑i=1N(t(i)z(i))−∑i=1Nloge(1+e−z(i))

비용함수의 형태로 정의하면 다음과 같다. 이제부터는 이 비용함수를 w에 대해 미분하도록 하겠다.

−l(w)=∑i=1N(t(i)z(i))+∑i=1Nloge(1+e−z(i))

이제 wi에 대한 편미분을 통해 최적의 W 벡터를 찾으면 된다. 그런데 여기서 문제가 있다. w에 대해 미분해야 하는데, 식은 치환변수인 z로 작성되어 있다. 따라서 여기서는 연쇄법칙에 따른 매개변수 미분을 진행한다.

이때, xi의 차원수만큼 wi가 존재할 것이고, wi 하나를 찾기위해선 N개의 데이터에 대해 연산해야 한다. 즉, w1...wD의 미분계수가 존재하고, 이는 미분계수 벡터의 형태로 나타낼 수 있다.

▽W=[δδw1,δδw1,...,δδwD]T

이것을 이제 데이터 세트 하나씩 곱하여 미분하면 된다. 아마 벡터 연산으로 나타내면 아래와 같은 모양일 것이다. 모든 식에서 0이 나오도록 연립방정식을 풀면 된다.

우행렬의 열벡터 각각은 i번째의 x 벡터이다. 보통은 벡터연산으로 z=WTXi로 풀기에, 아래와 같이 행렬로 연산한다.

[δδw1δδw2...δδwD][t(1)w1x1(1)+log(1+e−w1x1(1))...t(N)w1x1(N)+log(1+e−w1x1(N))t(1)w2x2(1)+log(1+e−w2x2(1))...t(N)w2x2(N)+log(1+e−w2x2(N))...t(1)wDxD(1)+log(1+e−wDxD(1))...t(N)wDxD(N)+log(1+e−wDxD(N))]=▽WT⋅[t(1)WTX1+log(1+e−WTX1)...t(N)WTXN+log(1+e−WTXN)]

여기서 δδw1[t(1)w1x1(1)+log(1+e−w1x1(1))]의 식만 계산 해보자.

💡로그의 미분

case1 y=ln x

dydx=1x

case2 y=loga x

dydx=1x ln a

case3 y=ln kx

dydx=1x

이제부터는 특정 스칼라 값들에 대한 연산이므로 w1=w,x1(1)=x로 간단하게 표현하겠다.

자연로그가 있으므로, 자연로그 내부의 식은 u로 치환하여, 연쇄 법칙을 적용하여 미분해야 한다.

1. w에 대해 미분하기

σσwtwx+σσwln(u), u=1+e−wxtx+σσuln(u)σuσw

2. u에 대해 미분하기

σσuln(u)=1u

3. u를 w에 대해 미분하기

σuσw=−xe−wx

4. 대입하기

tx+(1+e−wx)−1(−xe−wx)=x(t−(1+e−wx)−1(e−wx))

5. 미분값이 0이 되는 지점의 w 구하기

  • t=1 인 경우
x+(1+e−wx)−1(−xe−wx)=x(1−(1+e−wx)−1(e−wx))=01−e−wx(1+e−wx)=0e−wx(1+e−wx)=1e−wx=1+e−wx−wx=ln(1+e−wx)w=−ln(1+e−wx)x

exponential의 미분은 매우 까다롭다. 그런데, 이런 exponential에 대한 미분을 NxD회 실행해야 하므로, 하나의 CPU를 이용해 순차적으로 미분방정식을 푸는 것은 매우 비효율적이고, 사실상 불가능한 일일 것이다.

따라서, 딥러닝의 학습 시에는 그래픽 연산에 최적화된 프로세서인 GPU를 이용해, dn에 대해 동시다발적으로 다수의 데이터에서 미분이 계산될 수 있도록 병렬연산을 진행한다. 즉, t1의 시간에 N개의 w1에 대한 미분 연산이 동시에 진행된다는 것이다. GPU 개수가 많으면 많을 수록, 동시에 계산할 수 있는 차원 수도 많아지므로 훨씬 빨라진다.

물론, 제아무리 GPU와 같은 하드웨어의 도움을 받아 연산을 효율화 할 수 있다고 하더라도, 기본적으로는 차원 수를 줄인다던가, 데이터 샘플링을 통해 대표성을 갖는 데이터세트에 대해서만 학습을 진행한다던가 하는 방법을 함께 사용한다.

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+ \ No newline at end of file diff --git a/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/README.html b/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/README.html index d359f3d..502f61e 100644 --- a/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/README.html +++ b/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/README.html @@ -12,7 +12,7 @@ - + @@ -47,7 +47,7 @@ - + @@ -63,7 +63,7 @@
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- + \ No newline at end of file diff --git a/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-1.html b/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-1.html index 160185e..e08112a 100644 --- a/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-1.html +++ b/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-1.html @@ -63,7 +63,7 @@
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1. 선형대수란? ​

선형대수란? ​

💡 대수학(algebra)이란, 일련의 공리들을 만족하는 수학적 구조들의 일반적인 성질을 연구하는 수학의 한 분야 이다. 즉, 대수학에서 numerric이 아닌, 문자로 일반화된 수학적 법칙을 다루는 학문 분야이다.

대수학이란 수식에 대한 일반화된 법칙을 다루는 분야로 관련된 단어로는 ‘알고리즘’이 있다. 알고리즘이라는 것은 ‘어떤 문제를 해결하는 방식’이라는 일반적인 정의가 있으며, 컴퓨터 공학에서는 알고리즘을 특정 프로그래밍 문법이나 시스템에 기반하지 않은 general한 문제 해결 프로세스를 의미하고 보통 의사 코드(pseudo-code)로 표현된다.

알고리즘이나 대수학 모두 어떠한 상황(도메인)에서도 바뀌지 않는 일반화된 문제해결 법칙을 의미한다는 점에서 일맥상통한다고 할 수 있다. 실제로 어원도 같다.

선형대수를 이해할 때 일차방정식을 생각하면 쉽다. 일차방정식은 해가 1개이거나, 해가 아예 없거나, 혹은 해가 무한대일 수 있다. 해가 무한대인 경우는 일차방정식의 항등식을 생각하면 된다. 항등식이란 두 변수에 어떠한 값을 넣어도 항상 그 값이 같은 식을 의미한다. (A = B)

y=ax1+bx2+cx3+...+mxm

선형대수에서는 변수값에 영향을 주는 인자, 그러니까 위 식에서 a,b, c, m과 같은 수를 매개변수(parameter) 라고 하고, 변화하는 값을 변수(variable) 이라고 한다. 여기서 변수는 x1, x2, .. 그리고 y이다.

스칼라, 벡터, 매트릭스, 텐서 … ​

스칼라(Scalar) ​

스칼라는 0차원의 값으로 변수 하나, 값 한 개를 생각하면 된다.

x=3

위 식에서 x를 스칼라 값이라고 한다.

벡터(vector) ​

벡터는 1차원의 값이다. 즉, 스칼라가 연속적으로 있는 행렬을 의미한다.

vector=[1,2,3,4]

벡터 내부의 스칼라 개수를 차원(dimension)이라고 한다. 일반적 의미에서의 점→선→면→입체→초입방체를 얘기할 때의 그 차원이다. 따라서 벡터로 공간 상 한 지점에 값을 표시하는 것이 가능하다.

행렬 (Matrics) ​

벡터를 쌓아 행,열의 2차원 평면으로 나타낸 것을 행렬이라고 한다. 행렬은 벡터가 중첩된 형태로 표현된다.

matrics=[[1,2,3,4],[5,6,7,8],[9,10,11,12]]

텐서(Tensor) ​

머신러닝 분야에서 도입한 개념이다. N차원만큼 벡터를 중첩시킨 것을 의미한다. 보통은 3차원 이상부터 ‘텐서’ 명칭을 붙인다.

tensor=[[[1,2,3],[4,5,6]][[1,2,3][4,5,6]]]

파이토치, 텐서플로우 ​

파이토치와 텐서플로우 모두 GPT를 활용해 텐서 연산을 지원하는 라이브러리이다. 텐서플로우가 텐서 연산 라이브러리의 거의 시초격이고, 그 다음에 파이토치가 나왔다. 그래서 커뮤니티도 텐서플로우 쪽이 좀 더 크긴 한데 요즘은 파이토치도 많이 쓴다. 두 라이브러리 모두 Numpy와의 호환성이 뛰어나다.

구현 자체는 텐서플로우의 경우 C++, 파이토치의 경우 파이썬으로 구현되었다. 그래서 그런지, 텐서플로우는 연산결과가 C++이고 이를 파이썬으로 다시 보여줘서 좀 지저분하게 나온다. 반대로 파이토치는 텐서플로우보다는 결과가 깔끔하게 나온다.

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- + \ No newline at end of file diff --git a/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-2.html b/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-2.html index 5417bb3..830ba02 100644 --- a/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-2.html +++ b/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-2.html @@ -63,7 +63,7 @@
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2. 벡터, 행렬 기본개념 ​

전치행렬 ​

전치행렬이란, 행렬에서 행과 열의 값을 전치시킨 행렬이다. 스칼라값의 현재 인덱스 값의 행, 열값이 서로 바뀌는 것을 의미한다. 전치 행렬임을 나타낼 때는 보통 윗첨자로 T를 기입한다. 벡터가 전치되는 경우, 수식적으로는 그냥 벡터의 스칼라 값들을 세로로 기입하면 되지만, 파이썬이나 기타 프로그래밍 언어에서는 ‘세로로 기입’한다는 개념 자체가 없기에, 굳이 전치 벡터를 나타내고 싶다면 벡터가 아닌 행렬로 값을 표현하여야 제대로 적용된다.

Transposedmatrics=matricsT

예시 ​

mat=[[1,2,3,4,5]]matT=[[1],[2],[3],[4]]

Norm ​

Norm이란, 벡터의 크기를 수치화 하는 함수이다. 기본적으로 Norm 이라 하면 L2 Norm으로 받아들인다. 이 외에 L1 Norm, 제곱 L2, Max Norm이 있다. Max Norm 을 제외한 Norm 함수에 대한 일반식 LP Norm은 다음과 같다.

||X||p=(∑i|xi|)1p

L2 Norm ​

가장 일반적으로 사용되는 Norm이다. ||X|| 만 되어 있으면 그냥 L2 Norm으로 보면 된다.

||X||2=∑ixi2

L2 Norm이라는 것은 결국 0 벡터(원점벡터)에서 해당 벡터까지의 유클리드 거리와 같다.

L1 Norm ​

L1 Norm은 제곱값에 루트를 씌우는 것이 아닌, 단순히 절댓값을 더한 값이다. 보통 0인 것과 0이 아닌 것을 가려야 하는 경우에 사용한다.

||X||1=∑i|xi|

Max Norm ​

max norm은 단순히 벡터의 스칼라 중 절댓값이 가장 큰 값을 취하는 norm이다.

||X||=MAXi|xi|

벡터의 종류 ​

단위벡터(unit vector) ​

단위 벡터란 Norm이 1인 벡터로, 이때 Norm은 L1이 될 수도, L2가 될 수도 있다. 보통은 L2 Norm으로 단위 벡터를 정하면, 원점에서 유클리드 거리가 1인 벡터이다.

단위 벡터를 기하학적으로 이해하면, 크기가 1이나, 방향성을 유지하고 있는 벡터이다. 즉, 어떤 벡터를 특정 방향으로 바꿔주고자 한다면, 그 방향의 단위 벡터를 곱하여 바꿔줄 수 있다.

기저(basis) ​

기저란 벡터공간에서 선형 관계에 있지 않는 벡터들의 모음으로, 서로 독립인 벡터들이다. 어떤 공간을 이루는 필수적인 구성요소이다.

기저가 성립하는 조건은 다음과 같다.

1. spanned set이 vector space의 전제가 되어야 한다

즉, 모든 벡터들은 선형 결합으로 나타날 수 있어야 하고

2. 기저 요소들은 서로 선형독립이어야 한다

이 두 조건을 만족하는 벡터들의 집합을 기저벡터라고 한다.

  • (+)선형독립과 선형종속

    • 선형 독립
      • 선형 독립이란 벡터의 개수만큼 차원이 확장(span)하는 것. 즉, 벡터와 벡터가 겹치지 않음. 쉽게 생각하면 두 벡터가 겹치지만 않으면 모두 독립이라고 할 수 있다.
    • 선형 종속
      • 선형 종속이란 벡터의 개수만큼 차원이 확장되지 못하는 경우를 의미함. 이 경우, 벡터가 다른 벡터와 겹치게 되어 발생함. 어떤 벡터를 나머지 벡터의 조합으로 나타낼 수 있으면 선형 종속이다.

  • 선형 결합(linear combination)

    벡터에 스칼라 값을 곱해 더한다는 것을 의미한다. 선형 결합의 결과를 span이라고 한다. span은 점, 선, 면 또는 다차원 공간일 수 있다.

  • (+)생성공간(span)

    생성공간이란 주어진 두 벡터의 조합으로 만들 수 있는 모든 가능한 벡터의 집합을 의미한다. 즉, 주어진 벡터들의 선형 조합을 통해 만들 수 있는 공간을 의미한다.

    마치 쫄쫄이 옷(기저)을 쭉 늘렸을 때, 늘어난 옷과 같은 상태를 span이라고 한다. 즉, 현재 가진 벡터들로 표현할 수 있는 영역을 span이라고 한다.

    3차원 공간에서 벡터 2개를 span하면, 평면까지만 span할 수 있다.

  • 차원

    차원은 주어진 공간에서 모든 기저들이 같는 벡터 수를 의미한다.

  • Rank

    컬럼 공간의 차원을 의미한다.

  • 열공간, 행공간

    컬럼벡터가 나타낼 수 있는 범위를 의미한다.

    C(A) 또는 range(A) 로 표기한다.

    반대로 행공간은 행 벡터가 나타낼 수 있는 범위를 의미한다. R(A)로 표기한다.

직교벡터 ​

직교벡터란 벡터와 해당 벡터의 전치벡터의 곱, 다시 말해 내적이 0이 되는 벡터를 의미한다. 기저벡터는 직교벡터가 될 수 있다. 그렇다고 모든 기저 벡터가 직교 벡터는 아니다.

XTX=0orX⋅X=0

내적이 0 이라는 것은 기하학적인 의미가 있다.

먼저 내적이라는 것은 삼각함수로 표현할 수 있다.

a→⋅b→=|a→||b→|cosθ

cos 90은 0이다. 한 벡터가 원점 벡터이거나 (0,1) (2,0) 같은 기저 벡터가 아닌 이상, 벡터의 곱샘으로만 0이 나올 수는 없다. 그 말은 곧, 두 벡터의 사잇각이 90도가 되어야만 내적값이 0이 될 수 있다는 것이다. 기하학적으로 보면 두 벡터가 직교한다는 것은 곳, 두 벡터의 성질이 완전히 반대라는 것을 의미한다.

정규직교벡터 ​

정규직교 벡터는 직교벡터이면서 단위벡터인 벡터를 의미한다. 즉, 두 벡터의 norm이 1인 동시에 직교해야한다.

Ref ​

09 선형 독립(Linear independence), Span, 기저(Basis) 그리고 차원(Dimension) · linear algebra

[선대] 2-5강. 행렬의 곱셈과 네 가지 관점 (열공간 (column space) 등)

[선대] 2-6강. span 과 column space (열공간) 직관적 설명

[선대] 2-7강. 선형 독립과 기저 (linearly independent & basis) 직관적 설명

[행렬대수학] 전치행렬(Transposed Matrix)

09 선형 독립(Linear independence), Span, 기저(Basis) 그리고 차원(Dimension) · linear algebra

단위벡터, 기저벡터, span, rank, linear projections

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- + \ No newline at end of file diff --git a/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-3.html b/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-3.html index 9fc968e..4c3b643 100644 --- a/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-3.html +++ b/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-3.html @@ -63,7 +63,7 @@
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3. 텐서 연산 ​

상태: 완료 키워드: 벡터, 선형대수, 코드, 텐서, 행렬

텐서 전치 ​

스칼라, 벡터의 전치는 모두 텐서 전치의 일환이다. 전치를 기하학적으로 생각해보자면, 텐서의 크기를 유지시키면서 다른 차원으로 이동시키는 것이라고 할 수 있다.

스칼라 전치 ​

스칼라의 전치는 자기 자신이다. 즉, 스칼라는 방향이 없으므로 전치해도 형상의 변화가 없다

벡터 전치 ​

벡터 전치는 벡터의 행, 열값이 반대로 바뀌는 것이다. 쉽게 말해 행값, 열값이 반대로 바뀌는 것이다.

텐서 전치 ​

텐서의 전치는 텐서의 행/열의 대각 행렬(i=j)을 기준으로 값들이 반대로 바뀌는 것 을 의미한다.

(AB)T=BTAT

텐서의 스칼라 연산 ​

텐서에 스칼라를 더하거나 곱하는 연산은, 텐서의 형상을 유지시키며 텐서의 모든 값에 연산이 적용된다.

파이토치나 텐서플로우 모두 연산자 오버로딩을 통해, 마치 numeric한 값들을 연산하는 것처럼 기본 연산자(+,-..)를 사용해 계산할 수 있다.

아마다르곱 ​

아마다르곱 ​

아마다르곱은 형상이 동일한 두 벡터가 곱해지는 것을 의미한다. 아마다르곱은 동일한 인덱스값을 가진 원소들끼리 곱셈 연산을 진행하면 된다.

아마다르곱과 행렬곱은 다른 개념이다. 엄밀히 따지면, 아마다르곱은 행렬 곱에서 파생된 것인데, 형상이 동일한 행렬에 대한 곱셈 연산을 일반화 한 것이 아마다르곱이다.

아마다르곱은 일반적인 벡터의 곱(⋅)과는 다른 연산자( odot : ⊙)를 사용한다.

A⊙B

파이토치, 텐서플로우 모두 일반적인 행렬곱 코드를 사용해서 계산한다.

행렬곱 ​

행렬에 대한 곱셈은 벡터의 내적을 통해 이뤄진다. 행렬을 각각 행 벡터, 열 벡터로 쪼갠 후, 행,열 벡터에 대해 내적 한 값이 행렬곱이다. 결국엔 벡터에 대한 내적 연산이므로, 행 벡터로 쪼개지는 좌측 연산자의 열 개수와 열 벡터들로 쪼개지는 우측 연산자의 행 개수가 동일해야 한다.

img1

m∗n벡터와 n∗k 벡터의 곱셈연산의 결과 벡터는 m∗k 벡터이다.

일반적인 곱셈 연산과는 다르기에, 곱셈의 교환법칙은 성립하지 않으나 결합법칙, 분배법칙, 단위원 성질, 0의 곱셈성질은 동일하다.

행렬곱의 관점 ​

  1. 내적

    행렬곱을 두 행렬의 행, 열 벡터 각각의 내적으로 보는 관점이다. 행렬에서 각 요소들을 각각 행벡터, 열벡터로 보고, 이 행 벡터들과 열 벡터들의 각 내적을 구하는 방법으로, 일반적인 행렬곱 풀이에 적용되는 관점이다.

    img2

    기본적으로 벡터는 컬럼 기반이다. 위 그림에서 A, B는 행렬이 아니고 벡터들을 변수로 치환한 벡터이다. 그 과정에서, A는 전치 벡터들로 구성되었고, B는 일반 벡터(열벡터)로 구성되었다. 그리고 이 둘의 곱은, 전치 벡터와 일반 벡터의 내적으로 구해진다.

  2. rank-1 matrix 합

    rank란 어떤 행렬이 표현할 수 있는 차원을 의미한다. rank-1은 곧, 행렬이 1차원만을 표현할 수 있다는 것을 의미한다. 즉, 행렬의 모든 요소를 벡터 공간에 표현해도 일직선밖에 표현할 수 없다. rank-1행렬은 1차원 행벡터에 1차원 열벡터를 곱함으로써 얻을 수 있다.

    rank-1 matrics 합으로 행렬곱을 구하는 과정은, 행렬 요소를 각각 열벡터, 행벡터로 분해하여 곱셈 연산을 진행하고, 마지막에 연산 결과를 모두 더하는 과정으로 이뤄진다.

    A의 첫 번째 열벡터, B의 첫 번째 행벡터를 곱하면 행렬이 나온다. 이 계산 과정을 벡터 차원수만큼 반복해주고, 도출된 행렬들을 모두 더해주면 행렬곱 결과를 얻을 수 있다.

    ∑i=1nanbnTai∀n∈A,bi∀nT∈B

  3. column space 관점

    img1

    img2

    선형연립방정식 및 선형회귀에서 차용하는 관점이다. 열벡터에 스칼라인 변수를 곱해 나오는 스팬 값으로 변수의 해를 찾는 방식이다.

img3

  1. row-space 관점

    column-space 의 반대 관점이다. 이번에는 행 벡터에 변수를 곱해 나오는 행 스팬 값으로부터 변수의 해를 찾는 방식이다. 이때 현재 식에 전치 행렬을 적용하여, 변수 벡터를 행 벡터로 바꿔주고, 기존의 스칼라 열 벡터들도 행 벡터로 바꿔준다.

    주로 딥러닝 분야에서 취하는 관점이다. 행 벡터의 각 요소를 선형결합의 형태로 표현할 수 있기 때문이다. (리스트로 한번에 표현하여 쌓아버리는게 용이하기 때문)

    img4

벡터의 사영과 내적 ​

벡터의 내적 ​

A벡터와 B벡터의 내적 값은 곧 A 벡터와 B 벡터의 닮은 정도를 구하는 것이다. 여기서, 벡터는 방향값이라는 것이 존재하므로, A벡터의 B벡터에 대한 내적값과 B벡터의 A벡터에 대한 내적값은 전혀 다른 의미를 갖는다. 벡터 공간 상에서, A 벡터를 B 벡터 상에 표현했을 때의 크기와 B벡터를 A벡터 상에 표현하는 것은 완전히 다른 의미이기 때문이다. 이때 A를 B벡터 상에 나타내는 것을 사영(projection) 한다고 부른다.

벡터의 내적은 각 벡터의 크기와, 두 벡터의 사잇각의 곱으로 구할 수 있다.

a⋅b=||b||cosθ||a||

벡터의 크기 구하기 ​

|A|⋅BBTB

앞선 장에서 설명했듯이, 벡터의 크기는 벡터의 모든 요소의 제곱합에 제곱근을 취한 값이다.

|A|=∑i∈Aai2

또는, 다음과 같은 방식으로 나타낼 수도 있다.

||A||=ATA

단위벡터를 구하는 방식을 위의 공식을 활용해 나타내면 다음과 같다.

unitA=A||A||=AATA

정사영 ​

정사영은 어떤 점을 평면 위로 내린 수선의 발을 의미한다. 벡터에서의 정사영이란, 벡터 a, b가 존재할 때, 벡터b의 종점을 벡터a에 수선의 발을 내리고, 그것을 종점으로 하는 벡터를 의미한다.

삼각함수를 이용하면, 정사영 벡터의 크기를 구할 수 있다.

||projab||=||b||cosθ

하지만 $$\theta$$를 알기 어려운 경우가 대부분이다. 사잇각을 모를 때, 벡터 b의 종점과 a에 내린 수선의 발 까지의 거리가 최소라는 점을 이용해 식을 유도하는 방법이 있다.

  • 정사영 공식 유도

    정사영유도.jpg

θ값 없이 정사영을 구하는 공식은 다음과 같다.

projab=a(a⋅ba⋅a)

정사영과 내적의 관계 ​

벡터a의 단위벡터를 정사영 벡터의 크기를 구하는 식에 곱하면 정사영 벡터를 구할 수 있다.

projab=||b||cosθaa

즉, 유도한 공식(사잇각 모를때 구하는 방법)과 equation 하면

a(a⋅ba⋅a)=||b||cosθaa

이 식에서 a와 b의 내적을 유도할 수 있다.

a⋅ba⋅a=||b||cosθ||a||a⋅a=||a||2∴a⋅b||a||2=||b||cosθ||a||a⋅b=||b||cosθ||a||2||a||∴a⋅b=||b||cosθ||a||

python에서 내적 계산 방법 ​

numpy ​

python
np.dot(X,Y)

torch ​

python
torch.dot(torch.tensor([]), torch.tensor([]))

tensorflow ​

다른 라이브러리들과는 달리, 두 텐서를 곱한 다음 축소연산한다.

python
tf.reduce_sum(tf.multiply(X,Y))

Ref ​

[선대] 2-5강. 행렬의 곱셈과 네 가지 관점 (열공간 (column space) 등)

행렬 곱에 대한 또 다른 시각 - 공돌이의 수학정리노트 (Angelo's Math Notes)

벡터의 내적, 정사영으로 깊이 생각하기(개념편)

Copyright © 2024 전자두뇌만들기.

- + \ No newline at end of file diff --git a/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-4.html b/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-4.html index c033be2..5e4b86c 100644 --- a/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-4.html +++ b/docs/.vitepress/dist/contents/MATH/linear-algebra-basic/linear-algebra-basic-chap-4.html @@ -12,7 +12,7 @@ - + @@ -47,7 +47,7 @@ - + @@ -63,8 +63,8 @@
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4. 행렬 성질 ​

대칭행렬 (Symetric Matrix) ​

대칭 행렬은 자신의 전치 행렬이 원래의 자기 자신과 같은 행렬이다. 즉, AT=A 인 행렬을 의미한다.

대칭행렬을 이루기 위해선 원소의 row index와 column index가 반대로 바뀌어도 동일한 원소를 가져야 한다. aij==aji 라는 것이다.

대칭 행렬을 이루기 위한 또 다른 조건은 정방 행렬이어야 한다는 것이다. 즉, 행렬의 행 크기가 열 크기가 동일해야 한다.

여기서, i==j 인 원소들을 대각성분(diagonal entry) 이라고 한다. 대칭행렬은 대각성분을 기준으로 대칭을 이룬다.

대칭 행렬 예시

단위행렬(unit matrix, 또는 항등행렬(identity matrix)) ​

대각성분이 모두 1이고, 나머지 요소돌은 모두 0인 대칭 행렬을 의미한다. 대칭행렬이므로 정방행렬이다.

단위 행렬과 벡터의 곱은 벡터에 스칼라 1을 곱하는 것과 같다. 즉, 곱셈 연산 전후로 벡터 값에 변화가 없다.

단위행렬은 In으로 나타낸다. n은 행(열)의 개수를 의미한다. 벡터에 단위 연산을 곱하는 연산을 수식으로 나타내면 다음과 같다

Iw=w

앞선 챕터에서 다뤘던 행렬곱에서는 좌항의 행렬을 계수로, 우항의 벡터를 변수로 이해하였다.

이러한 관점에 따라 단위 행렬과 벡터의 곱을 이해하면 다음 그림과 같다.

단위행렬과 벡터의 곱

이러한 단위 행렬의 성질은 역행렬과 직교행렬의 특성을 설명할 때 다시 언급된다.

대각행렬(diagonal matrix) ​

대각행렬은 대각 성분을 제외한 원소들이 모두 0이라는 점에서는 동일하나, 대각성분이 1이 아닌 값도 될 수 있다. 즉, 단위 행렬은 대각 행렬의 일종이라고도 이해할 수 있다.

대각성분벡터(주대각선)로 대각행렬을 표기하는 방법은 다음과 같다. 이때 v는 주대각선을 이루는 대각성분벡터이다.

D=diag(v)

대각행렬의 대각성분을 이루는 벡터(diag(v))와 벡터의 곱셈연산은 아마다르곱(⊙)과 같다.

diag(v)X=v⊙X

역행렬(Inverse Matrix) ​

역행렬은 같은 꼴의 정방행렬 A와 단위행렬 I에 대해 AX=XA=I를 만족시키는 행렬을 의미한다. 일반적으로 행렬곱은 교환법칙이 성립하지 않아 곱셉 순서를 바꾸면 그 결과가 달라지나, 역행렬의 경우 곱셈의 순서를 바꾸어도 그 결과가 단위행렬로 항상 동일하다.

역행렬은 정방행렬이어야 하며, 어떠한 정방행렬에 있어 역행렬은 오직 하나만 존재한다.

모든 정방행렬이 역행렬을 가지는 것은 아니며, 이러한 행렬을 특이행렬(Singular Matrix) 이라고 칭한다. 역행렬을 갖는 행렬은 가역행렬(Non Singular Matrix)이라고 부르기도 한다.

역행렬에 대한 표기는 A−1로 한다. 분수의 표기와 동일하다. 단위행렬과 행렬의 곱이 스칼라 1을 행렬에 곱한 결과와 비슷하다는 점에서 분수가 연상되기도 한다.

역행렬을 본격적으로 이해해보기 전에, 헹렬식에 대해 알아보자.

행렬식 (determinant) ​

행렬식이란, 정방행렬에 하나의 수를 대응시키는 함수를 의미한다. 행렬을 통해 연립일차방정식의 해(크래머 공식)를 구하기 위해 고안되었다고 한다. 그 외에도 고윳값을 계산할 떄도 등장하는 용어다. 역행렬에서 행렬식에 대해 다루는 이유는, 행렬식을 통해 역행렬의 존재성을 판별하기 때문이다.

이차정사각행렬에서 기하학적으로 행렬식을 이해하면, 평면상에서 행렬의 열벡터를 표현했을때, 각각의 종점을 연장하여 평행사변형을 만들었을 때의 넓이가 행렬식의 값과 같다. 정사각행렬의 열벡터로 나타낸 평행사변형

삼차정사각행렬에서의 행렬식의 값도 마찬가지로, 3차원 공간상에 표현된 평행육면체의 부피와 같다. 2X2 정방행렬에서의 행렬식은 다음과 같다

det[abcd]=ad−bc

2X2 정방행렬에서의 행렬식 연산은 간단하다. 대각선을 이루는 원소들끼리 곱하고, 이들을 더하기만 하면 된다.

3X3 정방행렬에서의 행렬식 연산은 좀 복잡하다. 두 가지 방법으로 구해볼 수 있다.

전개 방식

행렬에서 마지막 열을 제외한 나머지 열을 마지막 열의 다음에 붙여준다

이 상태에서 다음 연산을 진행해준다.

3X3 행렬식 1

그 다음엔 반대로 구한다.

3X3 행렬식 2

마지막으로 앞서 구한 두 값을 뺀다.

3X3 행렬식 3

여인수와 소행렬 ​

앞서 짧게 소개한 전개 방식은 사실 여인수전개를 통한 계산을 간단히 한 것이다.

소행렬을 통한 계산을 위해선 소행렬과 여인수전개의 개념을 먼저 숙지해야 한다.

소행렬 (minor determinant)

소행렬이란, 특정 열과 행을 제거하고 만든 부분행렬에 대한 행렬식을 의미한다.

행렬에 제외되는 행, 열을 아래첨자로 표기하면 된다. 또는 소행렬에 절댓값 기호를 취해서 나타내는 방법도 있다.

위 식은 i행, j열의 원소들을 제하고 남은 부분에 대한 행렬식을 의미한다. 그림으로 나타내면 다음과 같다.

소행렬

여인수(cofactor)와 여인수전개

여인수전개는 여인수로 행렬식을 구하는 방법을 의미한다. 라플라스 전개라고 부르기도 한다.

여인수는 소행렬에 (-1)^(i+j)를 곱한 값을 의미한다. 식으로 나타내면 다음과 같다.

C=Cofator,Aij=SubmatrixC=(−1)(i+j)Aij

여인수 전개에서는 어떠한 행, 열에 대해 여인수 전개를 진행해도 동일한 값이 도출된다. 따라서, 한 행이나 열이 모두 동일한 값이거나 0이 많이 포함된 경우, 이 행/열에 대해 여인수 전개를 진행하면 매우 효율적으로 계산을 진행할 수 있다.

여인수전개는 4x4, 5x5 등 모든 차원의 정방행렬에 대해 적용될 수 있다.

다음은 3X3 행렬에서 여인수전개로 행렬식 값을 구한 과정을 나타낸 그림이다.

여인수 전개

위 그림대로 식들을 전개하면 다음의 식을 얻을 수 있다.

a21∗(−1)(2+1)⋅A21+a22∗(−1)(2+2)⋅A22+a23∗(−1)(2+3)⋅A23

실제 값을 대입하면 다음의 결과를 얻을 수 있다.

0∗(−1)∗(2∗7−5∗5)+0∗(1)∗(2∗5−1∗7)+0∗(−1)∗(1∗5−2∗2)=0

코드상으로 구현하면, 5X5 에서 4X4, 3X3 ... 으로 점차 줄어들어 쉽게 계산할 수 있는 2X2의 소행렬로 나눠 계산한 후, 다른 계산 결과와 합쳐 나가는 방식을 취할 것이다. 이처럼 자기 자신을 더 작게 나누어 계산 가능한 사이즈로 나눠 계산한 후 원래의 상태로 거슬러 올라가며 합쳐나가는 알고리즘은 재귀적 프로그래밍으로 구현했을 때 효과적이다. 하지만 대체로 텐서 연산 관련 라이브러리에 잘 구현되어 있다.

파이썬 상에서 행렬식은 구하고자 한다면, 복잡한 구현 없이 다음의 코드로 쉽게 구할 수 있다.

X = np.array([[1,2,4],[2,-1,3],[0,5,1]])
-np.linalg.det(X)

행렬식과 역행렬 ​

행렬식을 알아본 이유는, 행렬식을 통해 정방행렬에 역행렬이 존재하는지를 확인할 수 있기 때문이다.

역행렬이 존재하기 위해선, 행렬식의 값이 0이 아니어야 한다. 즉, det(A)≠0이 역행렬의 성립조건이다. 앞서 언급했듯이, 행렬식의 값은 벡터의 종점을 벡터의 크기만큼 연장시킨 평행사변형의 넓이이다. 행렬값이 0이라는 것은 다시 말하면, 두 벡터가 한 직선 상에 놓인 것이나 마찬가지이다.

역행렬을 구하는 수식은 다음과 같다.

A−1=1det(A)[C11C12...C1nC21C22...C2n........Cn1....Cnn]

직교형렬 (Orthogonal Matrix) ​

직교 행렬은 정방행렬이면서 행벡터, 열벡터가 모두 수직이어야 한다. 또한, 열벡터, 행벡터의 크기가 1이어야 한다.

즉, 직교행렬은 행렬의 열이 정규직교벡터(Orthogonal Vector)로 이뤄진 행렬을 의미한다.

또한, 직교행렬의 전치행렬과 직교 행렬의 곱은 단위행렬이다. 즉, QTQ=I 이다. QTQ=I이므로 역행렬이 곧 전치행렬 이기도 하다. 즉, QT=Q−1 이다.

직교행렬과 벡터의 곱의 크기는 벡터의 크기와 같다. 이는 아래의 식으로 증명 할 수 있다.

||Qv||=||v||||Qv||=(Qv)TQVvTQTQv=vTIvvTv=||v||

직교행렬과 직교행렬의 곱은 직교행렬이다. 이는 직교행렬이 각도와 길이, 내적을 보존하는 행렬이기 때문이다.

Ref ​

[선형대수학] Determinant / 행렬식

행렬의 성분, 두 행렬이 서로 같을 조건

행렬식 이 영상만 보면 기본 끝! | 행렬식의 성질 | 기본행연산과 행렬식

orthogonal matrix(직교행렬) 과 matrix of orthonormal columns

Copyright © 2024 전자두뇌만들기.

- +np.linalg.det(X)

행렬식과 역행렬 ​

행렬식을 알아본 이유는, 행렬식을 통해 정방행렬에 역행렬이 존재하는지를 확인할 수 있기 때문이다.

역행렬이 존재하기 위해선, 행렬식의 값이 0이 아니어야 한다. 즉, det(A)≠0이 역행렬의 성립조건이다. 앞서 언급했듯이, 행렬식의 값은 벡터의 종점을 벡터의 크기만큼 연장시킨 평행사변형의 넓이이다. 행렬값이 0이라는 것은 다시 말하면, 두 벡터가 한 직선 상에 놓인 것이나 마찬가지이다.

역행렬을 구하는 수식은 다음과 같다.

A−1=1det(A)[C11C12...C1nC21C22...C2n........Cn1....Cnn]

직교형렬 (Orthogonal Matrix) ​

직교 행렬은 정방행렬이면서 행벡터, 열벡터가 모두 수직이어야 한다. 또한, 열벡터, 행벡터의 크기가 1이어야 한다.

즉, 직교행렬은 행렬의 열이 정규직교벡터(Orthogonal Vector)로 이뤄진 행렬을 의미한다.

또한, 직교행렬의 전치행렬과 직교 행렬의 곱은 단위행렬이다. 즉, QTQ=I 이다. QTQ=I이므로 역행렬이 곧 전치행렬 이기도 하다. 즉, QT=Q−1 이다.

직교행렬과 벡터의 곱의 크기는 벡터의 크기와 같다. 이는 아래의 식으로 증명 할 수 있다.

||Qv||=||v||||Qv||=(Qv)TQVvTQTQv=vTIvvTv=||v||

직교행렬과 직교행렬의 곱은 직교행렬이다. 이는 직교행렬이 각도와 길이, 내적을 보존하는 행렬이기 때문이다.

Ref ​

[선형대수학] Determinant / 행렬식

행렬의 성분, 두 행렬이 서로 같을 조건

행렬식 이 영상만 보면 기본 끝! | 행렬식의 성질 | 기본행연산과 행렬식

orthogonal matrix(직교행렬) 과 matrix of orthonormal columns

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STEP 3. 깃허브 Actions 탭에서 디플로이 되고 있는지 확인 ​

이제 디플로이용 브랜치에서 npm run docs:build를 수행한 후 push 하면 디플로이가 자동으로 실행된다. 디플로이 진행상황은 깃허브 action 탭에서 확인하면 된다. 디플로이가 완료되면 아래 사진처럼 초록색 체크 아이콘이 뜬다.

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