normal matrix

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

properties:

• Equivalently a complex matrix $A\in {ℂ}^{n\times n}$ is said to be normal if it satisfies $[A,A^{\ast }]=0$ where $[,]$ is the commutator bracket.

• Equivalently a real matrix $A\in {ℝ}^{n\times n}$ is said to be normal if it satisfies $[A,A^T]=0$ where $[,]$ is the commutator bracket.

• 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 $A$.

• A complex square matrix is diagonal if and only if it is normal, triangular.(see theorem for normal triangular matrices).

examples:

• where $a,b\in ℝ$

see also:

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