**NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW
VERSION:** "Mathematics of the Discrete
Fourier Transform (DFT), with Audio Applications --- Second
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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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## Matrices

A

matrixis defined as a rectangular array of numbers, e.g.,

which is a (“two by two”) matrix. A general matrix may be , where is the number ofrows, and is the number ofcolumns. For example, the general matrix is

(Either square brackets or large parentheses may be used.) Theth element^{7.3}of a matrix may be denoted by or . The rows and columns of matrices are normally numbered from instead of from; thus, and . When , the matrix is said to besquare.The

transposeof a real matrix is denoted by and is defined by

Note that while is , its transpose is .A

complex matrix, is simply a matrix containing complex numbers. Thetransposeof a complex matrix is normally defined to includeconjugation. The conjugating transpose operation is called theHermitian transpose. To avoid confusion, in this tutorial, and the word “transpose” will always denote transpositionwithoutconjugation, while conjugating transposition will be denoted by and be called the “Hermitian transpose” or the “conjugate transpose.” Thus,

Example:The transpose of the general matrix is<!– MATH \begin{displaymath}## {\left[\begin{array}{cc} a & b \c & d \e & f \end{array}\right]}^{\hbox{\tiny T}}

\left[\begin{array}{ccc} a & c & e \b & d & f \end{array}\right] \end{displaymath} –>

while the conjugate transpose of the general matrix is

<!– MATH \begin{displaymath}## {\left[\begin{array}{cc} a & b \c & d \e & f \end{array}\right]}^{\ast }

\left[\begin{array}{ccc} \overline{a} & \overline{c} & \overline{e }

\overline{b} & \overline{d} & \overline{f }\end{array}\right] \end{displaymath} –>

is the special case of an matrix, and arow-vector

(as we have been using) is a matrix. In contexts where matrices are being used (only this section for this course), it is best to define all vectors ascolumn vectorsand to indicate row vectors using the transpose notation, as was done in the equation above.