PlanetMath: normal matrix

(more info)

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normal matrix

(Definition)

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 where denotes the transpose.

properties:

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.
Equivalently a real matrix $A \in \mathbb{R}^{n\times n}$ is said to be normal if it satisfies where is the commutator bracket.
Let be a square complex matrix of order . It follows from Schur's inequality that if is a normal matrix then $\sum_{i=1}^n \vert\lambda_i\vert^2 = \operatorname{trace} A^\ast A$ where $^\ast$ is the conjugate transpose and $\lambda_i$ are the eigenvalues of .
A complex square matrix is diagonal if and only if it is normal, triangular.(see theorem for normal triangular matrices).

examples:

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