normal matrix

A complex matrix $A\in\mathbb{C}^{n\times n}$ is said to be normal if $A^{\ast}A=AA^{\ast}$ where ${}^{\ast}$ denotes the conjugate transpose.
Similarly for a real matrix $A\in\mathbb{R}^{n\times n}$ is said to be normal if $A^{T}A=AA^{T}$ where $T$ denotes the transpose.

properties:

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Equivalently a complex matrix $A\in\mathbb{C}^{n\times n}$ is said to be normal if it satisfies $[A,A^{\ast}]=0$ where $[,]$ is the commutator bracket.
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Equivalently a real matrix $A\in\mathbb{R}^{n\times n}$ is said to be normal if it satisfies $[A,A^{T}]=0$ where $[,]$ is the commutator bracket.
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Let $A$ be a square complex matrix of order $n$ . It follows from Schur’s inequality that if $A$ is a normal matrix then $\sum_{i=1}^{n}|\lambda_{i}|^{2}=\operatorname{trace}A^{\ast}A$ where ${}^{\ast}$ is the conjugate transpose and $\lambda_{i}$ are the eigenvalues of $A$ .
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A complex square matrix is diagonal if and only if it is normal, triangular.(see theorem for normal triangular matrices).

examples:

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$\begin{pmatrix}a&b\\ -b&a\\ \end{pmatrix}$ where $a,b\in\mathbb{R}$
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$\begin{pmatrix}1&i\\ -i&1\\ \end{pmatrix}$

see also:

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Wikipedia, http://www.wikipedia.org/wiki/Normal_matrixnormal matrix

Title	normal matrix
Canonical name	NormalMatrix
Date of creation	2013-03-22 13:41:10
Last modified on	2013-03-22 13:41:10
Owner	Daume (40)
Last modified by	Daume (40)
Numerical id	12
Author	Daume (40)
Entry type	Definition
Classification	msc 15A21
Synonym	normal
Related topic	TheoremForNormalTriangularMatrices