Matrix Multiplication

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). Matrix Multiplication

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Matrix Multiplication

Let ${A}^{\ be a general $M\ matrix and let $B$ denote a general$L\ matrix. Denote the matrix product by $C={A}^{\ or $C={A}^{\. Then matrix multiplication is carried out by computing the inner product of every row of ${A}^{\ with every column of $B$. Let the$i$th row of ${A}^{\ be denoted by ${\, $i=1, 2,\, and the$j$th column of $B$ by $\, $j=1,2,\. Then the matrix product $C={A}^{\ is defined as

\

This definition can be extended to complex matrices by using a definition of inner product which does not conjugate its second argument.7.4

Examples:

<!– MATH \begin{displaymath} \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] \end{displaymath} –>\



<!– MATH \begin{displaymath} \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] \end{displaymath} –>\



<!– MATH \begin{displaymath} \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] \end{displaymath} –>\



\

An $M\ matrix $A$ can only be multiplied on the right by an$L\ matrix, where $N$ is any positive integer. An $L\matrix $A$ can only be multiplied on the left by a $M\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 $AB\ 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:

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The identity matrix is denoted by $I$ and is defined as

\

Identity matrices are always square. The $N\ identity matrix $I$, sometimes denoted as $I_N$, satisfies $A\ for every$M\ matrix $A$. Similarly, $I_M\, for every $M\matrix $A$.

As a special case, a matrix ${A}^{\ times a vector $\ produces a new vector $\ which consists of the inner product of every row of ${A}^{\ with$\

\

A matrix ${A}^{\ times a vector $\ defines a linear transformationof $\. In fact, every linear function of a vector $\ 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., ${A}^{\ (constant along alldiagonals).

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:

\

where the alternate transpose notation “$\” is defined to include complex conjugation so that the above result holds also for complex vectors. Using this result, we may rewrite the general matrix multiply as
\

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