6-1 Matrix and matrix multiplication

In this chapter we will work with matrix and matrix multiplication. Of course, just learning how to multiply the matrices together is not going to be the goal of this chapter. But to learn more powerful tools in linear algebra, we need to start from the basics.

Let’s look at two matrix multiplication with vectors $A\pmb{x} = \pmb{b}$ :

$$\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 3 \end{bmatrix}$$

$$\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}\begin{bmatrix} \color{red}{0} \\ \color{red}{1} \end{bmatrix} = \begin{bmatrix} \color{red}{2} \\ \color{red}{4} \end{bmatrix}$$

If we combine the vectors into a matrix, and do the same thing with both vector $\pmb{x}$ and vector $\pmb{b}$, we will get :

$$\begin{bmatrix} 1 & 2\\ 3 & 4 \end{bmatrix}\begin{bmatrix} 1 & \color{red}{0} \\ 0 & \color{red}{1} \end{bmatrix} = \begin{bmatrix} 1 & \color{red}{2} \\ 3 & \color{red}{4} \end{bmatrix}$$

So here you are! This is the way of multiplying matrices! Although we don’t know what it is means or what is it for yet, we can still see something out of it.

First, you can see that the “shape” of the matrix is quite important here. Just like when we do transformation, we cannot make a 3 by 3 transformation matrix to transform a 1d vector. To multiply two matrices, the matrices must have the certain “shape”.

If we think about matrix & matrix multiplication as the combination of many column vector transformations, then like transformation, the size of the input column vector must agree with the size of the transformation matrix, to be specific, the column number of the first matrix must be the same as the size of the input column vector. Which happens to be the row number of the second matrix.

But how about the row number of the first matrix and the column number of the second matrix? Are they useful? Well yes!, they are just happen to be the dimension of the result matrix! The row number of the first matrix became the dimension of the output vector and the column number of the second matrix tells you how many of them are combining together to form the final matrix.

All of the details above seem complicated and hard to memorize, but here is one good thing about math, you don’t have to memorize them at all! Just write down any matrix and start multiplying them and combining the vectors. You will figure out all the details by yourself! Also luckily, we hardly work with non-square matrices, and for square matrices, rules are pretty simple: $n \times n$ matrix can only multiply the square matrix with the same size ($n \times n$) and the result will be another $n \times n$ matrix.

6-1 Matrix and matrix multiplication

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