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 m*n size matrix. Let B be an n*k 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 $b=\left[\begin{array}{c}8,9\\ 7,2\\ 6,1\end{array}\right]$

$$\left[\begin{array}{cc}2\cdot 8+3\cdot 7+5\cdot 6& 2\cdot 9+3\cdot 2+5\cdot 1\\ & \\ 1\cdot 8+4\cdot 7+7\cdot 6& 1\cdot 9+4\cdot 2+7\cdot 1\\ & \\ 0\cdot 8+1\cdot 7+8\cdot 6& 0\cdot 9+1\cdot 2+8\cdot 1\\ & \end{array}\right]=\left[\begin{array}{cc}67& 29\\ & \\ 78& 24\\ & \\ 55& 10\\ & \end{array}\right]$$

example

$$A=\left[\begin{array}{c}1,3,2\end{array}\right]B=\left[\begin{array}{c}[5],[6],[4]\end{array}\right]$$

Compute AB and BA

$$AB=\left[\begin{array}{c}1\cdot 5+3\cdot 6+2\cdot 4\end{array}\right]=\left[\begin{array}{c}31\end{array}\right]$$

—

BA is 3x3 b/c they’re 3x1 and 1x3 and we do m*k size

$$\left[\begin{array}{ccc}3\cdot 1& 5\cdot 3& 5\cdot 2\\ & & \\ 6\cdot 1& 6\cdot 3& 6\cdot 2\\ & & \\ 4\cdot 1& 4\cdot 3& 4\cdot 2\\ & & \end{array}\right]=\left[\begin{array}{ccc}5& 15& 10\\ & & \\ 6& 18& 12\\ & & \\ 4& 12& 8\\ & & \end{array}\right]$$