Matrix Multiplication

Part of Linear Algebra.

dot product

If x=(x_1, x_2, …, x_n), y = y_1,y_2,…,y_n) the dot product of x and y is the product

$x.y=x_1{y}_1{x}_2{y}_2...+x_n{y}_n$

marix multiplication let A be an mn size matrix. Let B be an nk matrix. The product matrix

AB is the matrix whose ij entry is the dot product of Ai . Bj

The resulting size will be an m*k matrix.

example

Compute AB if $a=\left[\begin{array}{c}2,3,5\\ 1,4,7\\ 0,1,8\end{array}\right]$ and $b=\left[\begin{array}{c}8,9\\ 7,2\\ 6,1\end{array}\right]$

= \begin{bmatrix} 67 & 29 \\ 78 & 24 \\ 55 & 10 \\ \end{bmatrix} $$ ### example $$ A= \begin{bmatrix}1,3,2\end{bmatrix} B=\begin{bmatrix}[5], [6], [4]\end{bmatrix} $$ Compute AB and BA $$ AB = \begin{bmatrix} 1 \cdot 5+3 \cdot 6+2 \cdot 4\end{bmatrix} = \begin{bmatrix}31\end{bmatrix} $$ -- BA is 3x3 b/c they're 3x1 and 1x3 and we do m*k size $$ \begin{bmatrix} 3 \cdot 1 & 5 \cdot 3 & 5 \cdot 2 \\ 6 \cdot 1 & 6 \cdot 3 & 6 \cdot 2 \\ 4 \cdot 1 & 4 \cdot 3 & 4 \cdot 2 \\ \end{bmatrix} = \begin{bmatrix} 5 & 15 & 10 \\ 6 & 18 & 12 \\ 4 & 12 & 8 \\ \end{bmatrix} $$