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Mathematics of the DFT
    Matrices

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Matrices

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... ...\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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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