Matrix Multiplication

Let $\mathbf{A}^{\!\hbox{\tiny T}}$ be a general $M\times L$ matrix and let $\mathbf{B}$ denote a general $L\times N$ matrix. Denote the matrix product by $\mathbf{C}=\mathbf{A}^{\!\hbox{\tiny T}}\, \mathbf{B}$ . Then matrix multiplication is carried out by computing the inner product of every row of $\mathbf{A}^{\!\hbox{\tiny T}}$ with every column of $\mathbf{B}$ . Let the $i$ th row of $\mathbf{A}^{\!\hbox{\tiny T}}$ be denoted by $\underline{a}^{\hbox{\tiny T}}_i$ , $i=1, 2,\ldots,M$ , and the $j$ th column of $\mathbf{B}$ by $\underline{b}_j$ , $j=1,2,\ldots,N$ . Then the matrix product $\mathbf{C}=\mathbf{A}^{\!\hbox{\tiny T}}\, \mathbf{B}$ is defined as

$$\mathbf{C}= \mathbf{A}^{\!\hbox{\tiny T}}\, \mathbf{B}= \left[\begin{array}{cccc} <\underline{a}^{\hbox{\tiny T}}_1,\underline{b}_1> & <\underline{a}^{\hbox{\tiny T}}_1,\underline{b}_2> & \cdots & <\underline{a}^{\hbox{\tiny T}}_1,\underline{b}_N> \\ <\underline{a}^{\hbox{\tiny T}}_2,\underline{b}_1> & <\underline{a}^{\hbox{\tiny T}}_2,\underline{b}_2> & \cdots & <\underline{a}^{\hbox{\tiny T}}_2,\underline{b}_N> \\ \vdots & \vdots & \cdots & \vdots \\ <\underline{a}^{\hbox{\tiny T}}_M,\underline{b}_1> & <\underline{a}^{\hbox{\tiny T}}_M,\underline{b}_2> & \cdots & <\underline{a}^{\hbox{\tiny T}}_M,\underline{b}_N> \end{array}\right].$$

This definition can be extended to complex matrices by using a definition of inner product which does not conjugate its second argument.^H.2

Examples:

$$\left[\begin{array}{cc} a & b \\ c & d \\ e & f \end{array}\right] \cdot \left[\begin{array}{cc} \alpha & \beta \\ \gamma & \delta \end{array}\right] = \left[\begin{array}{cc} a\alpha+b\gamma & a\beta+b\delta \\ c\alpha+d\gamma & c\beta+d\delta \\ e\alpha+f\gamma & e\beta+f\delta \end{array}\right]$$

$$\left[\begin{array}{cc} \alpha & \beta \\ \gamma & \delta \end{array}\right] \cdot \left[\begin{array}{ccc} a & c & e \\ b & d & f \end{array}\right] = \left[\begin{array}{ccc} \alpha a + \beta b & \alpha c + \beta d & \alpha e + \beta f \\ \gamma a + \delta b & \gamma c + \delta d & \gamma e + \delta f \end{array}\right]$$

$$\left[\begin{array}{c} \alpha \\ \beta \end{array}\right] \cdot \left[\begin{array}{ccc} a & b & c \end{array}\right] = \left[\begin{array}{ccc} \alpha a & \alpha b & \alpha c \\ \beta a & \beta b & \beta c \end{array}\right]$$

$$\left[\begin{array}{ccc} a & b & c \end{array}\right] \cdot \left[\begin{array}{c} \alpha \\ \beta \\ \gamma \end{array}\right] = a \alpha + b \beta + c \gamma$$

An $M\times L$ matrix $\mathbf{A}$ can be multiplied on the right by an $L\times N$ matrix, where $N$ is any positive integer. An $L\times N$ matrix $\mathbf{A}$ can be multiplied on the left by a $M\times L$ matrix, where $M$ is any positive integer. Thus, the number of columns in the matrix on the left must equal the number of rows in the matrix on the right.

Matrix multiplication is non-commutative, in general. That is, normally $\mathbf{A}\,\mathbf{B}\neq \mathbf{B}\,\mathbf{A}$ even when both products are defined (such as when the matrices are square.)

The transpose of a matrix product is the product of the transposes in reverse order:

$$(\mathbf{A}\mathbf{B})^{\hbox{\tiny T}} = \mathbf{B}^{\hbox{\tiny T}} \mathbf{A}^{\!\hbox{\tiny T}}$$

The identity matrix is denoted by $\mathbf{I}$ and is defined as

$$\mathbf{I}\isdef \left[\begin{array}{ccccc} 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \cdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end{array}\right]$$

Identity matrices are always square. The $N\times N$ identity matrix $\mathbf{I}$ , sometimes denoted as $\mathbf{I}_N$ , satisfies $\mathbf{A}\cdot \mathbf{I}_N =\mathbf{A}$ for every $M\times N$ matrix $\mathbf{A}$ . Similarly, $\mathbf{I}_M\cdot \mathbf{A}=\mathbf{A}$ , for every $M\times N$ matrix $\mathbf{A}$ .

As a special case, a matrix $\mathbf{A}^{\!\hbox{\tiny T}}$ times a vector $\underline{x}$ produces a new vector $\underline{y}= \mathbf{A}^{\!\hbox{\tiny T}}\underline{x}$ which consists of the inner product of every row of $\mathbf{A}^{\!\hbox{\tiny T}}$ with $\underline{x}$

$$\mathbf{A}^{\!\hbox{\tiny T}}\underline{x}= \left[\begin{array}{c} <\underline{a}^{\hbox{\tiny T}}_1,\underline{x}> \\ <\underline{a}^{\hbox{\tiny T}}_2,\underline{x}> \\ \vdots \\ <\underline{a}^{\hbox{\tiny T}}_M,\underline{x}> \end{array}\right].$$

A matrix $\mathbf{A}^{\!\hbox{\tiny T}}$ times a vector $\underline{x}$ defines a linear transformation of $\underline{x}$ . In fact, every linear function of a vector $\underline{x}$ can be expressed as a matrix multiply. In particular, every linear filtering operation can be expressed as a matrix multiply applied to the input signal. As a special case, every linear, time-invariant (LTI) filtering operation can be expressed as a matrix multiply in which the matrix is Toeplitz, i.e., $\mathbf{A}^{\!\hbox{\tiny T}}[i,j] = \mathbf{A}^{\!\hbox{\tiny T}}[i-j]$ (constant along diagonals).

As a further special case, a row vector on the left may be multiplied by a column vector on the right to form a single inner product:

$$\underline{y}^{\ast }{\underline{x}} = \langle \underline{x},\underline{y}\rangle % \ip brackets huge due to y-underbar$$