# 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:

• 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 $a,b\in\mathbb{R}$

• $\begin{pmatrix}1&i\\ -i&1\\ \end{pmatrix}$