# Matrix Multiplication

## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. Matrix Multiplication

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition", by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8. - Copyright © 2017-09-28 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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## Matrix Multiplication

Let be a general matrix and let denote a general matrix. Denote the matrix product by or . Then matrix multiplication is carried out by computing the inner product of every row of with every column of . Let theth row of be denoted by , , and theth column of by , . Then the matrix product 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 matrix can only be multiplied on the right by an matrix, where is any positive integer. An matrix can only be multiplied on the left by a matrix, where 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 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:

The identity matrix is denoted by and is defined as

Identity matrices are always square. The identity matrix , sometimes denoted as , satisfies for every matrix . Similarly, , for every matrix .

As a special case, a matrix times a vector produces a new vector which consists of the inner product of every row of with

A matrix 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., (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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