Sign in

Not a member? | Forgot your Password?

Search Online Books

Search tips

Free Online Books

Free PDF Downloads

A Quadrature Signals Tutorial: Complex, But Not Complicated

Understanding the 'Phasing Method' of Single Sideband Demodulation

Complex Digital Signal Processing in Telecommunications

Introduction to Sound Processing

C++ Tutorial

Introduction of C Programming for DSP Applications

Fixed-Point Arithmetic: An Introduction

Cascaded Integrator-Comb (CIC) Filter Introduction


FIR Filter Design Software

See Also

Embedded SystemsFPGA
Chapter Contents:

Search Mathematics of the DFT


Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?



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]}.

Previous: Round-Off Error Variance
Next: Matrix Multiplication

Order a Hardcopy of Mathematics of the DFT

About the Author: 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 (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 for details.


No comments yet for this page

Add a Comment
You need to login before you can post a comment (best way to prevent spam). ( Not a member? )