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 $A^TA=AA^T$ where $T$ 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 $[A,A^T]=0$ 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 $\sum_{i=1}^n |\lambda_i|^2 = \operatorname{trace} A^\ast A$ 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:

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

see also:

• Wikipedia, normal matrix