Matrix Multiplication

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Here's a small tutorial on how to perform matrix multiplication.

Dot Product

Before we begin let's first review how we can perform a dot product.

Given 2 vectors $\vec{a}$ , and $\vec{b}$ , we can find the dot product $\vec{a} \cdot \vec{b}$ with the following:

\begin{align} \vec{a} = \begin{bmatrix} a_x & a_y & a_z \end{bmatrix}\\ \vec{b} = \begin{bmatrix} b_x & b_y & b_z \end{bmatrix}\\ \\ \vec{a} \cdot \vec{b} = a_xb_x + a_yb_y + a_zb_z \end{align}

Notice that the dot product is a scalar value.

For example:

\begin{align} \vec{a} = \begin{bmatrix} 8 & 10 & 4 \end{bmatrix} \\ \vec{b} = \begin{bmatrix} 2 & 3 & 9 \end{bmatrix} \\ \\ \vec{a} \cdot \vec{b} &= (8 * 2) + (10 * 3) + (4 * 9)\\ &= 16 + 30 + 36\\ &= 82 \end{align}

We can use this technique to perform matrix multiplication.

Matrix Multiplication

Given two matrices $\boldsymbol{A}$ , and $\boldsymbol{B}$ , they can be multiplied to a matrix, $\boldsymbol{C}$ if the first matrix's number of columns is the same as the second matrix's number of rows.

This means that if we have a matrix dimensions of $m \times n$ , it can be mulitplied by a second matrix if it has dimensions of $n \times p$ .

For example given a matrix $\boldsymbol{A}$ with dimensions of $3 \times 3$ , and matrix $\boldsymbol{B}$ of dimensions $3 \times 4$ , it satisfies the conditions that the dimensions of the matrices are valid for matrix multiplication.

\begin{equation} \boldsymbol{A} = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix} \boldsymbol{B} = \begin{bmatrix} j & k & l & m \\ o & p & q & r \\ s & t & u & v \end{bmatrix} \end{equation}

We can then visualize the matrix multiplication of $\boldsymbol{A} * \boldsymbol{B}$ as taking every row in $\boldsymbol{A}$ , as a vector $\vec{v_i}$ , and finding the dot product of every column in $\boldsymbol{B}$ , as a column vector $\vec{u_i}$ .

\begin{equation} \vec{v_0} = \begin{bmatrix} a & b & c \end{bmatrix} \\ \vec{u_0} = \begin{bmatrix} j \\ o \\ s \end{bmatrix} \\ \\ \vec{v_i} \cdot \vec{u_i} = (a * j) + (b * o) + (c * s) \end{equation}

This will be the single element in the matrix $\boldsymbol{C}$ located at position $(0, 0)$ . Then, to find $\boldsymbol{C_{0,1}}$ , it will just be:

\begin{align} \vec{v_0} \cdot \vec{u_1} &= \begin{bmatrix} a & b & c \end{bmatrix} * \begin{bmatrix} k \\ p \\ t \end{bmatrix} \\ &= (a * k) + (b * p) + (c * t) \end{align}

We can then repeat the process until we have completed matrix $\boldsymbol{C}$ .

Other Notes

Matrix multiplication is anti-commutative. So, the order matters here in evaluation. Multiplying matrices $\boldsymbol{A}$ , and $\boldsymbol{B}$ as $\boldsymbol{A} * \boldsymbol{B}$ is not the same as $\boldsymbol{B} * \boldsymbol{A}$ .