matrix-arithmetic.Rmd

# Matrix Arithmetic Functions

## Addition

$$
C = A + B
$$

### Derivatives {-}

$$
\frac{\partial}{\partial A} C = 1
\qquad
\frac{\partial}{\partial B} C = 1
$$

### Tangent {-}

$$
\dot{C} = \dot{A} + \dot{B}
$$

### Adjoints {-}

$$
\overline{A} \ \ {\small +}{=} \ \ \overline{C}
\qquad
\overline{B} \ \ {\small +}{=} \ \ \overline{C}
$$

## Subtraction

$$
C = A - B
$$

### Derivatives {-}

$$
\frac{\partial}{\partial A} C = 1
\qquad
\frac{\partial}{\partial B} C = -1
$$

### Tangent {-}

$$
\dot{C} = \dot{A} - \dot{B}
$$

### Adjoints {-}

$$
\overline{A} \ \ {\small +}{=} \ \ \overline{C}
\qquad
\overline{B} \ \ {\small +}{=} \ \ -\overline{C}
$$

## Multiplication

$$
C = A \cdot B
$$

### Derivatives {-}

$$
\frac{\partial}{\partial A} C = B
\qquad
\frac{\partial}{\partial B} C = A
$$

### Tangent {-}

$$
\dot{C} = \dot{A} \cdot B + A \cdot \dot{B}
$$

### Adjoints {-}

$$
\overline{A} \ \ {\small +}{=} \ \ \overline{C} \cdot B^{\top}
\qquad
\overline{B} \ \ {\small +}{=} \ \ A^{\top} \cdot \overline{C}
$$

## Negation

$$
C = -A
$$

### Derivatives {-}

$$
\frac{\partial}{\partial A} C = -1
$$

### Tangent {-}

$$
\dot{C} = -\dot{A}
$$

### Adjoint {-}

$$
\overline{A} \ \ {\small +}{=} \ \ -\overline{C}
$$

## Inverse

$$
C = A^{-1}
$$

### Tangent {-}

$$
\dot{C} = -C \cdot \dot{A} \cdot C
$$

### Adjoint {-}

$$
\overline{A} \ \ {\small +}{=} \ \ -C^T \cdot \overline{C} \cdot C^T
$$

## Determinant

$$
C = \det{A}
$$

### Tangent {-}

$$
\dot{C} = C \cdot \textrm{Tr} \left( A^{-1} \dot{A}\right)
$$

### Adjoint {-}

$$
\overline{A} \ \ {\small +}{=} \ \ \overline{C} \cdot C \cdot \left( A^{-1} \right)^T
$$

## Matrix left division

$$
C = A \char`\\ B = A^{-1} \cdot B
$$

### Tangent {-}

$$
\dot{C} = A^{-1} \left( \dot{B} - \dot{A} \cdot C \right)
$$

### Adjoint {-}

$$
\overline{A} \ \ {\small +}{=} \ \ -\overline{B} \cdot C^T
\qquad
\overline{B} \ \ {\small +}{=} \ \ \left( A^{-1} \right)^T \cdot \overline{C}
$$

## Matrix right division

$$
C = A / B = A \cdot B^{-1}
$$

### Tangent {-}

$$
\dot{C} = \left( \dot{A} - C \cdot \dot{B} \right) B^{-1}
$$

### Adjoint {-}

$$
\overline{A} \ \ {\small +}{=} \ \ \overline{C} \cdot \left( B^{-1} \right)^T
\qquad
\overline{B} \ \ {\small +}{=} \ \ -C^T \cdot \overline{A}
$$

## First quadratic form

$$
C = B^T \cdot A \cdot B
$$

### Tangent {-}

$$
\dot{C} = \dot{B^T} \cdot A \cdot B + B^T \cdot \dot{A} \cdot B + B^T \cdot A \cdot \dot{B}
$$

### Adjoint {-}

$$
\overline{A} \ \ {\small +}{=} \ \ B \cdot \overline{C} \cdot B^T
\qquad
\overline{B} \ \ {\small +}{=} \ \ A \cdot B \cdot \overline{C}^T + A^T \cdot B \cdot \overline{C}
$$

## Second quadratic form

$$
C = B^T \cdot A^{-1} \cdot B
$$

### Tangent {-}

$$
\dot{C} = \dot{B^T} \cdot A^{-1} \cdot B + B^T \cdot A^{-1} \cdot \dot{A} \cdot A^{-1} \cdot B + B^T \cdot A^{-1} \cdot \dot{B}
$$

### Adjoint {-}

$$
\overline{A} \ \ {\small +}{=} \ \ -\left( A^{-1} \right)^T \cdot B \cdot \overline{C} \cdot B^T \cdot \left( A^{-1} \right)^T
\qquad
\overline{B} \ \ {\small +}{=} \ \ A^{-1} \cdot B \cdot \overline{C}^T + \left( A^{-1} \right)^T \cdot B \cdot \overline{C}
$$