normal matrix
A complex matrix $A\in {\u2102}^{n\times n}$ is said to be normal if ${A}^{\ast}A=A{A}^{\ast}$ where ${}^{\ast}$ denotes the conjugate transpose^{}.
Similarly for a real matrix $A\in {\mathbb{R}}^{n\times n}$ is said to be normal if ${A}^{T}A=A{A}^{T}$ where $T$ denotes the transpose^{}.
properties:

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Equivalently a complex matrix $A\in {\u2102}^{n\times n}$ 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 ${\sum}_{i=1}^{n}{{\lambda}_{i}}^{2}=\mathrm{trace}{A}^{\ast}A$ 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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$\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill b\hfill & \hfill a\hfill \end{array}\right)$ where $a,b\in \mathbb{R}$

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$\left(\begin{array}{cc}\hfill 1\hfill & \hfill i\hfill \\ \hfill i\hfill & \hfill 1\hfill \end{array}\right)$
see also:

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

Canonical name  NormalMatrix 
Date of creation  20130322 13:41:10 
Last modified on  20130322 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 