4. Matrix Multiplication

2026-02-11

Matrix–vector Multiplication

When we are multiplying a $m \times n$ matrix $A$ by a vector $\vec{x}$ , we can write as below:

A\vec{x} = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n}\\ a_{2,1} & a_{2,2} & \cdots & a_{2,n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix} a_{1,1}x_1 + a_{1,2}x_2 + \cdots + a_{1,n}x_n\\ a_{2,1}x_1 + a_{2,2}x_2 + \cdots + a_{2,n}x_n\\ \vdots\\ a_{m,1}x_1 + a_{m,2}x_2 + \cdots + a_{m,n}x_n \end{bmatrix} \\[6pt] \text{Or} \\[6pt] A\vec{x} = \begin{bmatrix} \vec{v_1} & \vec{v_2} & \cdots & \vec{v_n} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = x_1\vec{v_1} + x_2\vec{v_2} + \cdots + x_n\vec{v_n}

The result is a $m$ -dimensional vector, or a $m \times 1$ matrix.

Tips

\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6\end{bmatrix} \begin{bmatrix}7 \\ 8 \\ 9\end{bmatrix} = 7\begin{bmatrix}1 \\ 4\end{bmatrix} + 8\begin{bmatrix}2 \\ 5\end{bmatrix} + 9\begin{bmatrix}3 \\ 6\end{bmatrix} = \begin{bmatrix}58 \\ 139\end{bmatrix}

It's obvious that $\vec{x} = I_n\vec{x}$ , where $I_n$ is the $n \times n$ identity matrix.

Let $A=(a_{ik})_{m\times s},\; B=(b_{kj})_{s\times n},\; C=AB$ . We say the product $C$ is an $m\times n$ matrix, whose elements are

c_{ij}=\sum_{k=1}^{s} a_{ik}b_{kj}\quad \bigl(i=1,\dots,m;\; j=1,\dots,n\bigr).

An Example

\begin{aligned} &\begin{bmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \end{bmatrix}_{2\times 3} \begin{bmatrix} 7 & 8\\ 9 & 10\\ 11 & 12 \end{bmatrix}_{3\times 2} \\[6pt] &\quad= \begin{bmatrix} 1\cdot7+2\cdot9+3\cdot11 & 1\cdot8+2\cdot10+3\cdot12\\ 4\cdot7+5\cdot9+6\cdot11 & 4\cdot8+5\cdot10+6\cdot12 \end{bmatrix}_{2\times 2} \\[6pt] &\quad= \begin{bmatrix} 58 & 64\\ 139 & 154 \end{bmatrix} \end{aligned}

The product $AB$ is only defined when the number of columns of $A$ equals the number of rows of $B$ .

AB = A \begin{bmatrix} \vec{b_1} & \vec{b_2} & \cdots & \vec{b_n} \end{bmatrix} = \begin{bmatrix} A\vec{b_1} & A\vec{b_2} & \cdots & A\vec{b_n} \end{bmatrix}

We can define $A^k$ :

A^k = \underbrace{A \cdot A \cdot \ldots \cdot A}_k

The power of a square matrix is only defined for square matrices.

Specifically, we define $A^0$ as the identity matrix $I_n$ of the same size as $A$ .