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A matrix is defined as a rectangular array of numbers, e.g.,

$\displaystyle \mathbf{A}= \left[\begin{array}{cc} a & b \\ [2pt] c & d \end{array}\right]

which is a $ 2\times2$ (``two by two'') matrix. A general matrix may be $ M\times N$ , where $ M$ is the number of rows, and $ N$ is the number of columns of the matrix. For example, the general $ 3\times 2$ matrix is

$\displaystyle \left[\begin{array}{cc} a & b \\ c & d \\ e & f \end{array}\right].

Either square brackets or large parentheses may be used to delimit the matrix. The $ (i,j)$ th elementH.1 of a matrix $ \mathbf{A}$ may be denoted by $ \mathbf{A}[i,j]$ , $ \mathbf{A}(i,j)$ , or $ \mathbf{A}_{ij}$ . For example, $ \mathbf{A}[1,2]=b$ in the above two examples. The rows and columns of matrices are normally numbered from $ 1$ instead of from 0 ; thus, $ 1\leq i \leq M$ and $ 1\leq j \leq N$ . When $ N=M$ , the matrix is said to be square.

The transpose of a real matrix $ \mathbf{A}\in{\bf R}^{M\times N}$ is denoted by $ \mathbf{A}^{\!\hbox{\tiny T}}$ and is defined by

$\displaystyle \mathbf{A}^{\!\hbox{\tiny T}}[i,j] \isdef \mathbf{A}[j,i].

While $ \mathbf{A}$ is $ M\times N$ , its transpose is $ N\times M$ . We may say that the ``rows and columns are interchanged'' by the transpose operation, and transposition can be visualized as ``flipping'' the matrix about its main diagonal. For example,

$\displaystyle \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].

A complex matrix $ \mathbf{A}\in{\bf C}^{M\times N}$ , is simply a matrix containing complex numbers. The transpose of a complex matrix is normally defined to include conjugation. The conjugating transpose operation is called the Hermitian transpose. To avoid confusion, in this tutorial, $ \mathbf{A}^{\!\hbox{\tiny T}}$ and the word ``transpose'' will always denote transposition without conjugation, while conjugating transposition will be denoted by $ A^{\ast }$ and be called the ``Hermitian transpose'' or the ``conjugate transpose.'' Thus,

$\displaystyle A^{\ast }[i,j] \isdef \overline{\mathbf{A}[j,i]}.

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``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 © 2014-10-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University