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Example:
The Hermitian transpose of the general $ 3\times 2$ matrix is

$\displaystyle \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].
$

A column vector, e.g.,

$\displaystyle \underline{x}= \left[\begin{array}{c} x_0 \\ [2pt] x_1 \end{array}\right]
$

is the special case of an $ M\times 1$ matrix, while a row vector, e.g.,

$\displaystyle \underline{x}^{\hbox{\tiny T}} = [\, x_0\; x_1 \,]
$

(as we have been using) is a $ 1\times N$ matrix. It is often helpful to adopt the convention that all vectors written without the transpose notation are column vectors, so that all row vectors require the transpose notation, as in the equation above.


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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 © 2014-04-06 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA