PX153 - K2 - matrix operations

addition

let $A = (a_{i j})_{m \times n}$ and $B = (b_{i j})_{p \times q}$
$A = B$ if and only if: $(i) m = p, n = q$ , and $(i i) a_{i j} = b_{i j} \forall i, j$
for matrices with the same dimensions:

A + B = [\begin{matrix} a_{11} + b_{11} & \dots & a_{1 n} + b_{1 n} \\ ⋮ & ⋱ & ⋮ \\ a_{m 1} + b_{m 1} & \dots & a_{m n} + b_{m n} \end{matrix}]

λ A = [\begin{matrix} λ a_{11} & \dots & λ a_{1 n} \\ ⋮ & ⋱ & ⋮ \\ λ a_{m 1} & \dots & λ a_{m n} \end{matrix}]

let $A = (a_{i j})_{m \times n}$ and $B = (b_{i j})_{n \times q}$
- note: $c o l ._{A} = r o w_{B}$
- if $c o l_{A} \neq r o w_{B}$ , $A B$ is not defined
the product, $A B$ , is matrix $C = (c_{i j})_{m \times q}$ , where,

c_{i j} = a_{i 1} b_{i j} + a_{i 2} b_{2 j} + \dots = \sum_{k = 1}^{n} a_{i k} b_{k j}

c_{1 j} = [\begin{matrix} a_{i 1} & a_{i 2} & \dots & a_{i n} \end{matrix}] \times [\begin{matrix} b_{1 j} \\ b_{2 j} \\ ⋮ \\ b_{n j} \end{matrix}] = \sum_{k = 1}^{n} a_{i k} b_{k j}

dot product of the $i^{t h}$ row of $A$ with the $j^{t h}$ column of $B$
matrix multiplication of square matrices of order $n$ can result in $A B \neq B A$
matrix multiplication is not commutative
the commutator:

[A, B] \equiv A B - B A

A_{i \times n} B_{n \times 1} = (\sum_{k = 1}^{n} a_{k} b_{k})_{1 \times 1}

A_{n \times 1} B_{1 \times n} = [\begin{matrix} a_{1} b_{1} & \dots & a_{1} b_{n} \\ ⋮ & ⋱ & ⋮ \\ a_{n} b_{1} & \dots & a_{n} b_{n} \end{matrix}]

eg: $A = {[\begin{matrix} 1 & 2 \\ - 1 & 3 \end{matrix}]}_{2 \times 2}$ , and $B = {[\begin{matrix} 5 & - 3 \\ 0 & - 2 \end{matrix}]}_{2 \times 2}$
$$AB = \begin{bmatrix} 5+0 & -3-4 \ -5+0 & 3-6 \end{bmatrix} = \begin{bmatrix} 5 & -7 \ -5 & -3\end{bmatrix}$$
$$BA = \begin{bmatrix} 5+3 & 10-9 \ 0+2 & 0-6\end{bmatrix} = \begin{bmatrix}8 & 1 \ 2 & -6\end{bmatrix}$$
$$[A,B] = \begin{bmatrix}-3 & -8 \ -7 & 3 \end{bmatrix}$$
eg: $C = {[\begin{matrix} 2 & 1 & 4 \\ 0 & 3 & - 1 \end{matrix}]}_{2 \times 3}$ , and $A = {[\begin{matrix} 1 & 2 \\ - 1 & 3 \end{matrix}]}_{2 \times 2}$
$A C = {[\begin{matrix} 2 & 7 & 2 \\ - 2 & 8 & - 7 \end{matrix}]}_{2 \times 3}$
- $C A$ is not a valid product
- however, $C^{T} = {[\begin{matrix} 2 & 0 \\ 1 & 3 \\ 4 & - 1 \end{matrix}]}_{3 \times 2}$
$C^{T} A = {[\begin{matrix} 2 & 4 \\ - 2 & 11 \\ 5 & 5 \end{matrix}]}_{3 \times 2}$
eg: $A = [\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}]$
$A^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$
$A^{k} = 0_{2 \times 2}$
$A^{k} = 0_{n \times n}$ but $A \neq 0_{n \times n}$