Class2.ipynb

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   "source": [
    "# Linear algebra Crash Course (with Python) Class 2 \n",
    "\n",
    "# Chap 2: 3d linear transformation\n",
    "\n",
    "## Interactive figure\n",
    "How is the interactive figure possible? It's because of the 3d linear transformation."
   ]
  },
  {
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Using matplotlib backend: <object object at 0x000001EB9B8A7550>\n"
     ]
    }
   ],
   "source": [
    "matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "05270f99-caf3-4c56-96c6-be7a2d844c2c",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "matplotlib qt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "b33993a3-9840-4167-bd43-9798e2274cc3",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Using matplotlib backend: QtAgg\n"
     ]
    }
   ],
   "source": [
    "matplotlib"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e3dc9d6f-e7df-4f38-8f17-4aa8e5bbaf5c",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "2c46d553-c4f2-4dcf-8680-84e8a1b156ac",
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   "source": [
    "# Define the coordinates for the dinosaur shape\n",
    "x = [6,3,1,-1,-2,-3,-4,-5,-5,-2,-5,-4,-2,-1,0,-1,1,2,1,3,5,6]\n",
    "y = [4,1,2,5,5,4,4,3,2,2,1,0,1,0,-3,-4,-4,-3,-2,-1,1,4]\n",
    "# Create a figure and axis\n",
    "ax = plt.figure().add_subplot(projection='3d')\n",
    "\n",
    "# Plot the dinosaur shape\n",
    "ax.plot(x, y,zs=1, zdir='z', label='curve in (x, y)')  # Use 'green' for the line color\n",
    "# Set the limits of the plot\n",
    "ax.set_xlim(min(x) - 1, max(x) + 1)  # Adjust the x-axis limits\n",
    "ax.set_ylim(min(y) - 1, max(y) + 1)  # Adjust the y-axis limits\n",
    "ax.set_zlim( - 2, 2)\n",
    "plt.gca().set_aspect('equal') #get current axes and set the aspect ratio\n",
    "\n",
    "# Display the plot\n",
    "plt.show()"
   ]
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  {
   "cell_type": "code",
   "execution_count": 7,
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   "source": [
    "import numpy as np\n",
    "\n",
    "ax = plt.figure().add_subplot(projection='3d')\n",
    "\n",
    "# Make the grid\n",
    "x = np.zeros(3)\n",
    "y = np.zeros(3)\n",
    "z = np.zeros(3)\n",
    "\n",
    "\n",
    "# Make the direction data for the arrows\n",
    "u = [1,0,0]\n",
    "v = [0,1,0]\n",
    "w = [0,0,1]\n",
    "\n",
    "ax.quiver(x, y, z, u, v, w, length=1, normalize=False)\n",
    "ax.set_xlim(-1, 1)\n",
    "ax.set_ylim(-1, 1)\n",
    "ax.set_zlim(-1, 1)\n",
    "\n",
    "plt.show()"
   ]
  },
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   "execution_count": 11,
   "id": "d55a89af-2b73-4ede-831c-d5bf55457daf",
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    {
     "name": "stdout",
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     "text": [
      "(10, 10)\n"
     ]
    }
   ],
   "source": [
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(projection='3d')\n",
    "\n",
    "# Make data\n",
    "rows,cols =(10,10)\n",
    "\n",
    "x1 =np.tile( np.linspace(-2, 2 , cols),(rows,1))\n",
    "print(x1.shape)\n",
    "y1 = x1.T\n",
    "z1 = 2*x1 + y1\n",
    "\n",
    "# Plot the surface\n",
    "\n",
    "ax.plot_surface(x1, y1, z1)\n",
    "\n",
    "#ax.quiver(x, y, z, u, v, w, length=1, normalize=False)\n",
    "\n",
    "# Set an equal aspect ratio\n",
    "ax.set_aspect('equal')\n",
    "\n",
    "plt.show()"
   ]
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   "cell_type": "markdown",
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   "source": [
    "Plane is a bit boring, let's try to put the dinasour on the plane."
   ]
  },
  {
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   "source": [
    "# Define the coordinates for the dinosaur shape\n",
    "x =np.array([6,3,1,-1,-2,-3,-4,-5,-5,-2,-5,-4,-2,-1,0,-1,1,2,1,3,5,6])\n",
    "y = np.array([4,1,2,5,5,4,4,3,2,2,1,0,1,0,-3,-4,-4,-3,-2,-1,1,4])\n",
    "#!!!!!!!!!!!\n",
    "z = 2*x+y\n",
    "#!!!!!!!!!!!\n",
    "\n",
    "# Create a figure and axis\n",
    "ax = plt.figure().add_subplot(projection='3d')\n",
    "\n",
    "# Plot the dinosaur shape\n",
    "ax.plot(x, y,z, zdir='z', label='dinosaur in projection')  \n",
    "ax.plot(x, y,zs=0, zdir='z', color='green')  # Use 'green' for the line color\n",
    "\n",
    "# Set the limits of the plot\n",
    "ax.set_xlim(min(x) - 1, max(x) + 1)  # Adjust the x-axis limits\n",
    "ax.set_ylim(min(y) - 1, max(y) + 1)  # Adjust the y-axis limits\n",
    "ax.set_zlim( - 10, 10)\n",
    "plt.gca().set_aspect('equal') #get current axes and set the aspect ratio\n",
    "\n",
    "# Display the plot\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f77ccbe-8ab8-49eb-aff2-e7e501ea3f36",
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   "source": [
    "This looks a bit funny. The prolonged dinasour is just looked like a shadow of the original dinasour. Could you measure the angle between the projected plane and the x-y plane?"
   ]
  },
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   "cell_type": "code",
   "execution_count": 13,
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   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "\n",
    "# Define the function for a sphere\n",
    "def sphere(radius, v, h):\n",
    "    phi, theta = np.mgrid[0.0:2.0*np.pi:20j, 0.0:np.pi:10j]\n",
    "    x = radius*np.cos(phi)*np.sin(theta) + v\n",
    "    y = radius*np.sin(phi)*np.sin(theta) + h\n",
    "    z = radius*np.cos(theta)\n",
    "    return x, y, z\n",
    "\n",
    "# Draw the eyes\n",
    "x, y, z = sphere(0.2, 0.5, -0.5)\n",
    "ax.plot_surface(x, y, z, color='b')\n",
    "x, y, z = sphere(0.2, -0.5, -0.5)\n",
    "ax.plot_surface(x, y, z, color='b')\n",
    "\n",
    "# Draw the mouth\n",
    "x = np.linspace(-1, 0, 100) \n",
    "y = 0.5*np.sin(x*np.pi) - 1.2\n",
    "z = np.zeros_like(x)\n",
    "ax.plot(x +0.5, y, z, 'r')\n",
    "ax.set_aspect('equal')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a1aa6279-6faa-4005-8796-8d0b84f7f2dd",
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   "source": [
    "Once again we want to fill the 3-d space with parallelepipedon by the vector $\\mathbf{u}_1$,$\\mathbf{u}_2$,$\\mathbf{u}_3$. It is quite obivious that we could do that as long as $\\mathbf{u}_1$,$\\mathbf{u}_2$,$\\mathbf{u}_3$ are linear independent.\n",
    "\n",
    "\n",
    "Similiar to the 2d case, we could represent a vector in two different coordinate systems and use a matrix to connect them.\n",
    "\n",
    "$$\\begin{bmatrix}\\mathbf{u}_1 & \\mathbf{u}_2 & \\mathbf{u}_3  \\end{bmatrix}\\begin{bmatrix} x \\\\ y \\\\ z \\end{bmatrix}_{\\mathbf{u}_1,\\mathbf{u}_2,\\mathbf{u}_3 }= \\begin{bmatrix} \\alpha \\\\ \\beta \\\\ \\gamma \\end{bmatrix}_{\\mathbf{e}_1,\\mathbf{e}_2,\\mathbf{e}_3 }$$\n",
    "which naturally relate to the equation.\n",
    "\n",
    "$$\\mathbf{M}\\mathbf{x}=\\vec{\\mathbf{\\alpha}}$$\n",
    "\n",
    "where $$\\mathbf{M}= \\begin{bmatrix}\\mathbf{u}_1 & \\mathbf{u}_2 & \\mathbf{u}_3 \\end{bmatrix}$$ \n",
    "\n",
    "On the other hand, you can see $\\mathbf{M}$ a linear map sending $\\mathbf{e}_1$ to $\\mathbf{u}_1$, sending $\\mathbf{e}_2$ to $\\mathbf{u}_2$,  sending $\\mathbf{e}_3$ to $\\mathbf{u}_3$. In this way, we can stay in the same cartesian coordinates. Both views have their pros and cons, and be ready to swtich if necessary.\n"
   ]
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   "cell_type": "markdown",
   "id": "e3b0408c-b553-4abd-84b9-399b1077a14f",
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   "source": [
    "***exercise: $\\mathbf{v}_1,\\mathbf{v}_2,\\mathbf{v}_3$ are 3-dimensional column vectors. Let $\\mathbf{V}=\\begin{bmatrix}\\mathbf{v}_1 & \\mathbf{v}_2 & \\mathbf{v}_3 \\end{bmatrix}$ and $\\mathbf{D}=\\begin{bmatrix}d_1 & 0 & 0 \\\\0 & d_2 & 0 \\\\0 &0 & d_3 \\end{bmatrix}$. Compute $\\mathbf{V}\\mathbf{D}$ and $\\mathbf{D} \\mathbf{V}^{\\top}$ in terms of $\\mathbf{v}_1,\\mathbf{v}_2,\\mathbf{v}_3$***"
   ]
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   "cell_type": "markdown",
   "id": "26be9f6a-27a4-4bd1-98f5-4c6035ab1a37",
   "metadata": {},
   "source": [
    "***练习：我们有三个三维列向量 $\\mathbf{v}_1,\\mathbf{v}_2,\\mathbf{v}_3$,将其并排在一起得到一个3x3的矩阵 $\\mathbf{V}$，另外我们有3x3对角线矩阵 $\\mathbf{D}$ 。计算 $\\mathbf{V}\\mathbf{D}$与 $\\mathbf{D} \\mathbf{V}^{\\top}$,并用 $\\mathbf{v}_1,\\mathbf{v}_2,\\mathbf{v}_3$ 表示其计算结果。***"
   ]
  },
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   "cell_type": "markdown",
   "id": "0cea5960-5533-4a22-881c-7195ce0f799e",
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   "source": [
    "***exercise: compute the following,***  \n",
    "(1).$\\begin{pmatrix}3&-2&1 \\\\ 1 &-1&2\\end{pmatrix}\\begin{pmatrix}-1&5\\\\-2&4\\\\3&-1 \\end{pmatrix}\n",
    "$\n",
    "(2).$\\begin{pmatrix}3&1&2&-1 \\\\ 0 &3&1&0 \\\\1&0&1&0 \\end{pmatrix}\\begin{pmatrix}0&5\\\\2&0\\\\0&-1\\\\-3&0 \\end{pmatrix}\n",
    "$\n",
    "(3).$\\begin{pmatrix}1 \\\\ 2 \\\\ 3 \\\\ 4\\end{pmatrix}\\begin{pmatrix}4&3&2&1 \\end{pmatrix}\n",
    "$  \n",
    "(4).$\\begin{pmatrix}4&3&2&1 \\end{pmatrix}\\begin{pmatrix}1 \\\\ 2 \\\\ 3 \\\\ 4\\end{pmatrix}\n",
    "$\n",
    "(5).$\\begin{pmatrix}0&-1&0 \\\\ 1 &0&1 \\\\ 0 &1&0 \\end{pmatrix}^4\n",
    "$\n",
    "(6).$\\begin{pmatrix}2&1&0 \\\\ 0 &2&1 \\\\ 0 &0&2 \\end{pmatrix}^4\n",
    "$\n",
    "(7).$\\begin{pmatrix}a&0&0 \\\\ 0&b&0 \\\\ 0 &0&c \\end{pmatrix}^n\n",
    "$\n",
    "(8).$\\begin{pmatrix}1&1 \\\\ 1 &1 \\end{pmatrix}^n\n",
    "$"
   ]
  },
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   "id": "8d0bb951-16c3-407f-934a-1d9b7c8111ee",
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   "source": [
    "2d linear transformation is a degenerated case for 3d linear transformation as you could embeded a 2d linear transformation into 3d linear transformation in this way.\n",
    "\n",
    "$$\\mathbf{M}=\\begin{bmatrix}a & b & 0 \\\\c & d & 0 \\\\0 &0 &0 \\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fd5d65ea-5b6c-4589-b7b1-a3dcfacb7219",
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   "source": [
    "### Determinant\n",
    "\n",
    "The determinant of a 3 by 3 matrix $\\mathbf{M}=\\begin{bmatrix}\\mathbf{u}_1&\\mathbf{u}_2&\\mathbf{u}_3\\end{bmatrix}=\\begin{bmatrix}a & d & g \\\\b & e & h \\\\c &f &m \\end{bmatrix}$:\n",
    "\n",
    "$$\\mathtt{det}(\\mathbf{M})\\stackrel{\\triangle}= aem+bfg+cdh-afh-bdm-ceg$$\n",
    "\n",
    "$$\\begin{align*}\n",
    "\\mathtt{det}(\\mathbf{M}) &= aem+bfg+cdh-afh-bdm-ceg \\\\\n",
    " &=a(em -fh) +b(fg-dm)+c(dh-eg) \\\\\n",
    " &=a \\cdot\\mathtt{det}(\\begin{bmatrix}e & f\\\\h&m \\end{bmatrix})+b \\cdot \\mathtt{det}(\\begin{bmatrix}f & d\\\\m&g \\end{bmatrix})+c \\cdot \\mathtt{det}(\\begin{bmatrix}d & e\\\\g&h \\end{bmatrix})\\\\\n",
    " &= \\begin{bmatrix}a\\\\b\\\\c \\end{bmatrix}\\cdot(\\begin{bmatrix}d\\\\e\\\\f \\end{bmatrix}\\wedge \\begin{bmatrix}g\\\\h\\\\m \\end{bmatrix} )= \\mathbf{u}_1 \\cdot(\\mathbf{u}_2 \\wedge \\mathbf{u}_3) \n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "1042341e-995f-424a-a69d-518aab5fc729",
   "metadata": {
    "tags": []
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   "source": [
    "import sympy as sp\n",
    "a,b,c,d,e,f,g,h,m = sp.symbols(\"a,b,c,d,e,f,g,h,m\",real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "97b63ecc-578c-469f-bf1f-4e5222adff09",
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a e m - a f h - b d m + b f g + c d h - c e g$"
      ],
      "text/plain": [
       "a*e*m - a*f*h - b*d*m + b*f*g + c*d*h - c*e*g"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M=sp.Matrix([[a,b,c],[d,e,f],[g,h,m]])\n",
    "M.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "cd02a474-82b3-408f-beeb-70a0831e8572",
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a e m - a f h - b d m + b f g + c d h - c e g$"
      ],
      "text/plain": [
       "a*e*m - a*f*h - b*d*m + b*f*g + c*d*h - c*e*g"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.T.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "4770ff10-b29d-465a-9f9d-d76231935731",
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   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "a*(e*m - f*h) + b*(-d*m + f*g) + c*(d*h - e*g)\n",
      "a*(e*m - f*h) + b*(-d*m + f*g) + c*(d*h - e*g)\n"
     ]
    }
   ],
   "source": [
    "from sympy.vector import CoordSys3D\n",
    "N = CoordSys3D('N')\n",
    "\n",
    "u1= a*N.i+b*N.j + c*N.k\n",
    "u2= d*N.i+e*N.j + f*N.k\n",
    "u3= g*N.i+h*N.j + m*N.k\n",
    "print(u1 & (u2.cross(u3)))\n",
    "print(u1.dot(u2.cross(u3)))            "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fccfb920-40c4-4486-945a-79d3884e0827",
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   "source": [
    "### Determinant and Volume\n",
    "In 3 dimensional world, $\\mathbf{u}_2 \\wedge \\mathbf{u}_3$ is a vector perpendicular to the plane spanned by $\\mathbf{u}_2 $ and $ \\mathbf{u}_3$ and has the length of $\\mathtt{sin} \\theta |\\mathbf{u}_2||\\mathbf{u}_3|$ ={ area of parallelogram made by the vectors: $\\vec{\\mathbf{0}},\\mathbf{u}_2 ,\\mathbf{u}_3,\\mathbf{u}_2 +\\mathbf{u}_3 $}. Therefore by the definition of volume, $\\mathtt{det}(M)=\\mathbf{u}_1 \\cdot(\\mathbf{u}_2 \\wedge \\mathbf{u}_3)  = $ volume of parallelepipedon made by the vectors: $\\vec{\\mathbf{0}},\\mathbf{u}_1 ,\\mathbf{u}_2,\\mathbf{u}_3,\\mathbf{u}_1 +\\mathbf{u}_2,\\mathbf{u}_1 +\\mathbf{u}_3,\\mathbf{u}_2 +\\mathbf{u}_3,\\mathbf{u}_1 +\\mathbf{u}_2+\\mathbf{u}_3 $.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dc688c0e-811f-467f-ad40-2a98590176df",
   "metadata": {},
   "source": [
    "**exercise: $\\mathbf{M}$  is a 3 by 3 matrix. Prove the following: if and only if $det(\\mathbf{M})=0$, there exist a non-zero 3 dimensional vector $\\mathbf{v}$ such that $\\mathbf{M}\\mathbf{v}=0$**"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2d0ec330-0006-453c-8b5f-f3aff039dc41",
   "metadata": {
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   },
   "source": [
    "### Inverse Matrix\n",
    " $\\mathbf{N}_{ij}$ is the determinant of the 2 by 2  matrix obtained from $\\mathbf{M}$ by removing the 𝑖-th row and the 𝑗-th column.  $\\mathbf{C}_{ij} = (-1)^{i+j} \\mathbf{N}_{ij}$ \n",
    "\n",
    "\n",
    "$$\\mathbf{M}^{-1}= \\frac{1}{\\mathtt{det}(\\mathbf{M})}\\begin{bmatrix}\\mathbf{C}_{11}&\\mathbf{C}_{12}&\\mathbf{C}_{13}\\\\ \\mathbf{C}_{21}&\\mathbf{C}_{22}&\\mathbf{C}_{23}\\\\ \\mathbf{C}_{31}&\\mathbf{C}_{32}&\\mathbf{C}_{33} \\end{bmatrix}^{\\top}$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "id": "1b608a3b-1637-45ef-aad3-69809b79e376",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}e m - f h & - b m + c h & b f - c e\\\\- d m + f g & a m - c g & - a f + c d\\\\d h - e g & - a h + b g & a e - b d\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[ e*m - f*h, -b*m + c*h,  b*f - c*e],\n",
       "[-d*m + f*g,  a*m - c*g, -a*f + c*d],\n",
       "[ d*h - e*g, -a*h + b*g,  a*e - b*d]])"
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(M.det())*M.inv()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a72408d0-1348-4b0d-aeb0-213af6a955c9",
   "metadata": {
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   "source": [
    "### Charasteristic equation, Eigenvalue, and Eigenvector\n",
    "\n",
    "Charasteristic equation of $n$ by $n$ square matrix $\\mathbf{M}$ is defined as $\\mathtt{det}(\\mathbf{M}-\\lambda \\mathbf{I})=0$ as a equation of $\\lambda$. $\\mathbf{I}$ is the diagonal identity matrix.\n",
    "- It's obivious that the degree of the Charasteristic equation is equal to $n$.(with some mathematical induction on the determinants) With fundermental theorem of algebra, we know it has $n$ complex roots.\n",
    "\n",
    "Eigenvalue of $\\mathbf{M}$ is defined as the roots of its eigenfunction. We write it down as $\\{\\lambda_{i}\\}$. Because the existence of mutiple roots, $|\\{\\lambda_{i}\\}|\\leq n$.\n",
    "\n",
    "Since $\\mathtt{det}(\\mathbf{M}-\\lambda_i \\mathbf{I})=0$, in 2 or 3 dimension, we know for every $\\lambda_i$ there exists a eigenvector $\\mathbf{v}_i$ such that $(\\mathbf{M}-\\lambda_i \\mathbf{I})\\mathbf{v}_i=0$. The statement is also true for general $n$ and we will discuss it again later in this course.\n",
    "\n",
    "By linearity, $\\mathbf{M}\\mathbf{v}_i=\\lambda_i \\mathbf{I}\\mathbf{v}_i= \\lambda_i \\mathbf{v}_i$\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0545fae3-7138-4b34-8f12-1ee42a39cb4a",
   "metadata": {},
   "source": [
    "Collecting all the eigenvectors $\\{\\mathbf{v}_i\\}$ as $\\mathbf{V}=[\\mathbf{v}_1,\\dots,\\mathbf{v}_m]$, $m\\leq n$.\n",
    "\n",
    "We have $\\mathbf{MV}= \\mathbf{VD}$, $\\mathbf{D}$ is a diagonal matrix with diagonal terms as  $\\lambda_i$. \n",
    "\n",
    "If we are lucky, $n=m$ and $\\mathbf{V}$ is invertible. We have $\\mathbf{M}= \\mathbf{VDV}^{-1}$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "50643601-5dcd-44d1-8483-cd86727874d3",
   "metadata": {},
   "source": [
    "### Examples \n",
    "Let's try to compute several examples:\n",
    "\n",
    "First, let's consider $\\mathbf{M} =\\begin{bmatrix}1 & 0 & 0 \\\\0 & 1 & 0 \\\\0 &0 &1 \\end{bmatrix}$ \n",
    "\n",
    "Obiviously $\\mathtt{det}(\\mathbf{M}-\\lambda \\mathbf{I})=-(\\lambda -1)^3$. $\\lambda_1=1$, We can also see that $\\mathbf{M}\\mathbf{v}=\\mathbf{v} ,\\forall \\mathbf{u} \\in \\mathbb{R}^3$\n",
    "\n",
    "Next, let's consider $\\mathbf{M} =\\begin{bmatrix}a & 0 & 0 \\\\0 & b & 0 \\\\0 &0 &c \\end{bmatrix}$\n",
    ", then $\\mathtt{det}(\\mathbf{M}-\\lambda \\mathbf{I})=-(\\lambda -a)(\\lambda -b)(\\lambda -c)$. $\\lambda_1=a$,$\\lambda_2=b$,$\\lambda_3=c$, $\\mathbf{v}_1 = \\begin{bmatrix}1 \\\\0 \\\\0\\end{bmatrix}$,$\\mathbf{v}_2 = \\begin{bmatrix}0 \\\\1 \\\\0\\end{bmatrix}$,$\\mathbf{v}_3 = \\begin{bmatrix}0 \\\\0 \\\\1\\end{bmatrix}$ \n",
    "\n",
    "Now, let's consider $\\mathbf{M} =\\begin{bmatrix}0 & 0 & 0 \\\\1 & 0 & 0 \\\\0 &1 &0 \\end{bmatrix}$, then $\\mathtt{det}(\\mathbf{M}-\\lambda \\mathbf{I})=-\\lambda^3$. $\\lambda_1=0$, $\\mathbf{v}_1= \\begin{bmatrix}0 \\\\0 \\\\1\\end{bmatrix}$, and we don't have any other eigenvector.\n",
    "\n",
    "For roataional matrix:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "36f9d873-042a-482c-91b0-276b670a3a5f",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "alpha, beta,theta, x1,y1,lambda1 = sp.symbols('alpha, beta,theta, x1,y1,lambda1',real=True)\n",
    "A = sp.Matrix([[sp.cos(alpha),-sp.sin(alpha)],[sp.sin(alpha),sp.cos(alpha)]])\n",
    "B = sp.Matrix([[sp.cos(alpha),-sp.sin(alpha),0],[sp.sin(alpha),sp.cos(alpha),0],[0,0,1]])\n",
    "C = sp.Matrix([[1,0,0],[0,sp.cos(theta),-sp.sin(theta)],[0,sp.sin(theta),sp.cos(theta)]])\n",
    "\n",
    "v = sp.Matrix([x1,y1])\n",
    "A = A- lambda1 *sp.eye(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "ae83eacf-3d95-4022-bb60-331b9b636214",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\lambda_{1} + \\cos{\\left(\\alpha \\right)} & - \\sin{\\left(\\alpha \\right)}\\\\\\sin{\\left(\\alpha \\right)} & - \\lambda_{1} + \\cos{\\left(\\alpha \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-lambda1 + cos(alpha),           -sin(alpha)],\n",
       "[           sin(alpha), -lambda1 + cos(alpha)]])"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 163,
   "id": "e1e3b978-b430-4bfe-9884-e403cb51e095",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\lambda_{1}^{2} - 2 \\lambda_{1} \\cos{\\left(\\alpha \\right)} + \\sin^{2}{\\left(\\alpha \\right)} + \\cos^{2}{\\left(\\alpha \\right)}$"
      ],
      "text/plain": [
       "lambda1**2 - 2*lambda1*cos(alpha) + sin(alpha)**2 + cos(alpha)**2"
      ]
     },
     "execution_count": 163,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 164,
   "id": "6816738e-7eaf-4964-8d04-aa8fb928f90c",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}e^{- i \\alpha}\\\\e^{i \\alpha}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[exp(-I*alpha)],\n",
       "[ exp(I*alpha)]])"
      ]
     },
     "execution_count": 164,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.Matrix(sp.solve(sp.Matrix(A).det(),lambda1 ))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "id": "9c07a73f-dab0-4285-b796-f85b5c57dbea",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\alpha \\right)} & - \\sin{\\left(\\alpha \\right)} \\cos{\\left(\\theta \\right)} & \\sin{\\left(\\alpha \\right)} \\sin{\\left(\\theta \\right)}\\\\\\sin{\\left(\\alpha \\right)} & \\cos{\\left(\\alpha \\right)} \\cos{\\left(\\theta \\right)} & - \\sin{\\left(\\theta \\right)} \\cos{\\left(\\alpha \\right)}\\\\0 & \\sin{\\left(\\theta \\right)} & \\cos{\\left(\\theta \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[cos(alpha), -sin(alpha)*cos(theta),  sin(alpha)*sin(theta)],\n",
       "[sin(alpha),  cos(alpha)*cos(theta), -sin(theta)*cos(alpha)],\n",
       "[         0,             sin(theta),             cos(theta)]])"
      ]
     },
     "execution_count": 169,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.Matrix(B*C)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "id": "de58bcf8-223a-4e5b-a401-4629a9dc268a",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\lambda_{1}^{3} + \\lambda_{1}^{2} \\cos{\\left(\\alpha \\right)} \\cos{\\left(\\theta \\right)} + \\lambda_{1}^{2} \\cos{\\left(\\alpha \\right)} + \\lambda_{1}^{2} \\cos{\\left(\\theta \\right)} - \\lambda_{1} \\sin^{2}{\\left(\\alpha \\right)} \\cos{\\left(\\theta \\right)} - \\lambda_{1} \\sin^{2}{\\left(\\theta \\right)} \\cos{\\left(\\alpha \\right)} - \\lambda_{1} \\cos^{2}{\\left(\\alpha \\right)} \\cos{\\left(\\theta \\right)} - \\lambda_{1} \\cos{\\left(\\alpha \\right)} \\cos^{2}{\\left(\\theta \\right)} - \\lambda_{1} \\cos{\\left(\\alpha \\right)} \\cos{\\left(\\theta \\right)} + \\sin^{2}{\\left(\\alpha \\right)} \\sin^{2}{\\left(\\theta \\right)} + \\sin^{2}{\\left(\\alpha \\right)} \\cos^{2}{\\left(\\theta \\right)} + \\sin^{2}{\\left(\\theta \\right)} \\cos^{2}{\\left(\\alpha \\right)} + \\cos^{2}{\\left(\\alpha \\right)} \\cos^{2}{\\left(\\theta \\right)}$"
      ],
      "text/plain": [
       "-lambda1**3 + lambda1**2*cos(alpha)*cos(theta) + lambda1**2*cos(alpha) + lambda1**2*cos(theta) - lambda1*sin(alpha)**2*cos(theta) - lambda1*sin(theta)**2*cos(alpha) - lambda1*cos(alpha)**2*cos(theta) - lambda1*cos(alpha)*cos(theta)**2 - lambda1*cos(alpha)*cos(theta) + sin(alpha)**2*sin(theta)**2 + sin(alpha)**2*cos(theta)**2 + sin(theta)**2*cos(alpha)**2 + cos(alpha)**2*cos(theta)**2"
      ]
     },
     "execution_count": 170,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(B*C- lambda1 *sp.eye(3)).det()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "id": "264ed4ce-428a-4113-bcbb-b70c7058bae0",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\lambda_{1}^{3} + \\lambda_{1}^{2} \\cos{\\left(\\alpha \\right)} \\cos{\\left(\\theta \\right)} + \\lambda_{1}^{2} \\cos{\\left(\\alpha \\right)} + \\lambda_{1}^{2} \\cos{\\left(\\theta \\right)} - \\lambda_{1} \\cos{\\left(\\alpha \\right)} \\cos{\\left(\\theta \\right)} - \\lambda_{1} \\cos{\\left(\\alpha \\right)} - \\lambda_{1} \\cos{\\left(\\theta \\right)} + 1$"
      ],
      "text/plain": [
       "-lambda1**3 + lambda1**2*cos(alpha)*cos(theta) + lambda1**2*cos(alpha) + lambda1**2*cos(theta) - lambda1*cos(alpha)*cos(theta) - lambda1*cos(alpha) - lambda1*cos(theta) + 1"
      ]
     },
     "execution_count": 171,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.simplify((B*C- lambda1 *sp.eye(3)).det())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "id": "3301eaae-7b38-4352-aa50-5fa0b066ee1a",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\lambda_{1}^{2} - \\lambda_{1} \\cos{\\left(\\alpha \\right)} \\cos{\\left(\\theta \\right)} - \\lambda_{1} \\cos{\\left(\\alpha \\right)} - \\lambda_{1} \\cos{\\left(\\theta \\right)} + \\lambda_{1} + 1$"
      ],
      "text/plain": [
       "lambda1**2 - lambda1*cos(alpha)*cos(theta) - lambda1*cos(alpha) - lambda1*cos(theta) + lambda1 + 1"
      ]
     },
     "execution_count": 172,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = sp.simplify((B*C- lambda1 *sp.eye(3)).det()/(1-lambda1))\n",
    "sp.simplify((B*C- lambda1 *sp.eye(3)).det()/(1-lambda1))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "36fdb387-5f52-4f3f-8306-02c935c9ab62",
   "metadata": {},
   "source": [
    "let $\\beta=-\\cos{\\left(\\alpha \\right)}\\cos{\\left(\\theta \\right)} -\\cos{\\left(\\alpha \\right)} - \\cos{\\left(\\theta \\right)} + 1$= $2-(1-\\cos{\\left(\\alpha \\right)})(1-\\cos{\\left(\\theta \\right)})$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "ed4cb773-90c2-4ee6-b0c3-8ad027ca8c77",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "\n",
    "fig = plt.figure()\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "\n",
    "alpha1, theta1 = np.mgrid[0.0:np.pi:10j, 0.0:np.pi:10j]\n",
    "\n",
    "z = 2-(1-np.cos(alpha1))*(1-np.cos(theta1))\n",
    "alpha1 = alpha1.flatten()\n",
    "theta1 = theta1.ravel()\n",
    "z= z.ravel()\n",
    "\n",
    "\n",
    "ax.plot(alpha1, theta1, z, 'r')\n",
    "ax.set_aspect('equal')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0412d898-e87e-4dbe-b453-f78902155491",
   "metadata": {},
   "source": [
    "***exercise: change the plot z = 2-(1-np.cos(alpha1))*(1-np.cos(theta1)), range:[0,pi]X[0,pi] into surface plot***"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "id": "59136eeb-cbdf-40f4-abe6-08a0cdc3a89b",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[-beta/2 - sqrt(beta**2 - 4)/2, -beta/2 + sqrt(beta**2 - 4)/2]"
      ]
     },
     "execution_count": 173,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.solve(lambda1**2 +beta*lambda1 +1,lambda1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c6e9470-8c13-418a-bfe3-d91d618231c2",
   "metadata": {},
   "source": [
    "Notice that $|\\beta|\\leq 2$, therefore $\\beta^2-4\\leq 0$. Therefore, by the formula of quadratic equation the roots of this equation is conjugated with norm 1."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bb19b68e-38c8-4fe5-89d7-62a21dac3639",
   "metadata": {},
   "source": [
    "Therefore, we gave the idea how to prove Euler’s fixed point theorem  which states that any motion of a rigid body on the surface of a sphere may be represented as a *rotation about an appropriately chosen rotation pole*, called an Euler pole. Geologists have used this theorem to understand the motions of tectonic plates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "id": "33ba99b8-174e-4762-a823-9c5cbe22caac",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "expr = (B*C- lambda1 *sp.eye(3)).det() /(lambda1-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "id": "0219e542-29f6-4d50-96cf-08055a6084a4",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1\\\\- \\frac{\\sqrt{\\sin^{2}{\\left(\\alpha \\right)} \\sin^{2}{\\left(\\theta \\right)} - 2 \\sin^{2}{\\left(\\alpha \\right)} \\cos{\\left(\\theta \\right)} - 2 \\sin^{2}{\\left(\\alpha \\right)} - 2 \\sin^{2}{\\left(\\theta \\right)} \\cos{\\left(\\alpha \\right)} - 2 \\sin^{2}{\\left(\\theta \\right)}}}{2} + \\frac{\\cos{\\left(\\alpha \\right)} \\cos{\\left(\\theta \\right)}}{2} + \\frac{\\cos{\\left(\\alpha \\right)}}{2} + \\frac{\\cos{\\left(\\theta \\right)}}{2} - \\frac{1}{2}\\\\\\frac{\\sqrt{\\sin^{2}{\\left(\\alpha \\right)} \\sin^{2}{\\left(\\theta \\right)} - 2 \\sin^{2}{\\left(\\alpha \\right)} \\cos{\\left(\\theta \\right)} - 2 \\sin^{2}{\\left(\\alpha \\right)} - 2 \\sin^{2}{\\left(\\theta \\right)} \\cos{\\left(\\alpha \\right)} - 2 \\sin^{2}{\\left(\\theta \\right)}}}{2} + \\frac{\\cos{\\left(\\alpha \\right)} \\cos{\\left(\\theta \\right)}}{2} + \\frac{\\cos{\\left(\\alpha \\right)}}{2} + \\frac{\\cos{\\left(\\theta \\right)}}{2} - \\frac{1}{2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[                                                                                                                                                                                               1],\n",
       "[-sqrt(sin(alpha)**2*sin(theta)**2 - 2*sin(alpha)**2*cos(theta) - 2*sin(alpha)**2 - 2*sin(theta)**2*cos(alpha) - 2*sin(theta)**2)/2 + cos(alpha)*cos(theta)/2 + cos(alpha)/2 + cos(theta)/2 - 1/2],\n",
       "[ sqrt(sin(alpha)**2*sin(theta)**2 - 2*sin(alpha)**2*cos(theta) - 2*sin(alpha)**2 - 2*sin(theta)**2*cos(alpha) - 2*sin(theta)**2)/2 + cos(alpha)*cos(theta)/2 + cos(alpha)/2 + cos(theta)/2 - 1/2]])"
      ]
     },
     "execution_count": 175,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.simplify(sp.Matrix(sp.solve(sp.simplify((B*C- lambda1 *sp.eye(3)).det()),lambda1 )))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5ce50974-5ecc-4385-a2cb-4aba45e5d997",
   "metadata": {},
   "source": [
    "***exercise compute the eigenfunction of ***"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "658187c2-f659-44b9-832e-19e3bc1d43d0",
   "metadata": {
    "tags": []
   },
   "source": [
    "### Orthogonal matrix\n",
    "\n",
    "We should elaborate a little bit more on orthogonal(orthonormal) matrices. The definition of  an orthogonal square matrix  $\\mathbf{U}$ is that $\\mathbf{U}\\mathbf{U}^\\top =\\mathbf{I}$. Roataional matrices are typical examples for orthgonal matrix. exercise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "b4c9f9fc-325f-4cef-9f61-ab6ccb961d92",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\cos{\\left(\\alpha \\right)} & - \\sin{\\left(\\alpha \\right)} & 0\\\\\\sin{\\left(\\alpha \\right)} & \\cos{\\left(\\alpha \\right)} & 0\\\\0 & 0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[cos(alpha), -sin(alpha), 0],\n",
       "[sin(alpha),  cos(alpha), 0],\n",
       "[         0,           0, 1]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c49bf1c1-b266-4815-a9db-68842dd30912",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\0 & \\cos{\\left(\\theta \\right)} & - \\sin{\\left(\\theta \\right)}\\\\0 & \\sin{\\left(\\theta \\right)} & \\cos{\\left(\\theta \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1,          0,           0],\n",
       "[0, cos(theta), -sin(theta)],\n",
       "[0, sin(theta),  cos(theta)]])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "04247a8f-305c-4b50-9ee9-3b6eeef6695e",
   "metadata": {
    "tags": []
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    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1, 0, 0],\n",
       "[0, 1, 0],\n",
       "[0, 0, 1]])"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.Matrix(sp.simplify(C*sp.transpose(C)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "id": "2fa94ab4-98a5-4ead-a6a8-816183c5284c",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1, 0, 0],\n",
       "[0, 1, 0],\n",
       "[0, 0, 1]])"
      ]
     },
     "execution_count": 178,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.Matrix(sp.simplify(B*C*sp.transpose(C)*sp.transpose(B)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 179,
   "id": "f54314e2-c38b-42c5-926e-620fb2c8d5b3",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[1, 0, 0],\n",
       "[0, 1, 0],\n",
       "[0, 0, 1]])"
      ]
     },
     "execution_count": 179,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.Matrix(sp.simplify(B*C*sp.transpose(B*C)))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "521186f9-5a5c-41f9-acf3-277ebc686bf2",
   "metadata": {},
   "source": [
    "#### Orthogonal matrix conserve length and angle\n",
    "\n",
    "It is obvious that given any 2 vectors with the same dimension, we have $\\langle \\mathbf{x},\\mathbf{y}\\rangle = \\mathbf{x}^\\top \\mathbf{y}$. Therefore for any square matrix $\\mathbf{M}$ with suitable dimensions(what are them?), $\\langle \\mathbf{x},\\mathbf{M}\\mathbf{y}\\rangle = \\mathbf{x}^\\top \\mathbf{M}\\mathbf{y}= \\mathbf{x}^\\top \\mathbf{M}^{\\top\\top}\\mathbf{y}= (\\mathbf{M}^{\\top}\\mathbf{x})^{\\top}\\mathbf{y}= \\langle \\mathbf{M}^{\\top}\\mathbf{x},\\mathbf{y}\\rangle$.\n",
    "\n",
    "Then for any orthogonal matrix $\\mathbf{U}$, $\\langle \\mathbf{U}\\mathbf{x},\\mathbf{U}\\mathbf{y}\\rangle =\\langle \\mathbf{U}^\\top\\mathbf{U}\\mathbf{x},\\mathbf{y}\\rangle = \\langle \\mathbf{x},\\mathbf{y}\\rangle = |\\mathbf{x}||\\mathbf{y}|\\mathtt{sin}\\theta$. Let $\\mathbf{x}=\\mathbf{y}$, we have $\\langle \\mathbf{U}\\mathbf{x},\\mathbf{U}\\mathbf{x}\\rangle= |\\mathbf{x}|^2$. Therefore we could say that orthogonal matrix is similiar to rigid motion which doesn't change their intrinsic relationship.\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "fa8dd858-2327-4446-9459-6b80fbc43f98",
   "metadata": {},
   "outputs": [],
   "source": [
    "*exercise?: prove that any orthogonal matrix is a combination of rotations and reflections*"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a0fef087-f6d3-4668-bc3d-508584bcc426",
   "metadata": {},
   "source": [
    "Conversely, if a finite dimensional matrix conserve length for all vector. Is it a orthogonal matrix?\n",
    "\n",
    "Suppose $\\langle \\mathbf{V}\\mathbf{x},\\mathbf{V}\\mathbf{x}\\rangle = \\langle \\mathbf{x},\\mathbf{x}\\rangle, \\forall \\mathbf{x}$, $\\langle \\mathbf{V}(\\mathbf{x}+\\mathbf{y}),\\mathbf{V}(\\mathbf{x}+\\mathbf{y})\\rangle = \\langle \\mathbf{V}\\mathbf{x},\\mathbf{V}\\mathbf{x}\\rangle +2\\langle \\mathbf{V}\\mathbf{x},\\mathbf{V}\\mathbf{y}\\rangle+\\langle \\mathbf{V}\\mathbf{y},\\mathbf{V}\\mathbf{y}\\rangle$\n",
    "$\\langle \\mathbf{x}+\\mathbf{y},\\mathbf{x}+\\mathbf{y}\\rangle = \\langle \\mathbf{x},\\mathbf{x}\\rangle +2\\langle \\mathbf{x},\\mathbf{y}\\rangle+ \\langle \\mathbf{y},\\mathbf{y}\\rangle$, $\\Rightarrow \\langle \\mathbf{x},\\mathbf{y}\\rangle =\\langle \\mathbf{V}\\mathbf{x},\\mathbf{V}\\mathbf{y}\\rangle,\\forall \\mathbf{x}, \\mathbf{y} \\Rightarrow \\langle \\mathbf{x},\\mathbf{y}\\rangle =\\langle \\mathbf{x},\\mathbf{V}^{\\top}\\mathbf{V}\\mathbf{y}\\rangle,\\forall \\mathbf{x}, \\mathbf{y}$\n",
    "With duality theory(check Lax textbook), we could view $\\langle \\cdot,\\mathbf{y}\\rangle$ as a induced linear function. Under the condition of finite dimensions, $\\mathbf{y} = \\mathbf{V}^{\\top}\\mathbf{V}\\mathbf{y}, \\forall \\mathbf{y}$, Therefore $\\mathbf{V}^{\\top}\\mathbf{V} = \\mathbf{I}$ and $\\mathbf{V}$ is orthogonal."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa9565c7-46e0-4c28-84d2-13ea1ae1782f",
   "metadata": {},
   "source": [
    "```\n",
    "The essence of man is no abstraction inherent in each single individual. In reality, it is the ensemble of the social relations.  --- \n",
    "```"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8e4a808a-6975-4329-ae01-046dca3b5c54",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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