**Search Mathematics of the DFT**

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

The *transpose* of a real matrix
is denoted by
and is defined by

A *complex matrix*
, 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,
and the word ``transpose'' will always denote transposition
*without* conjugation, while conjugating transposition will be
denoted by and be called the ``Hermitian transpose'' or the
``conjugate transpose.'' Thus,

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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