Normal matrix

Definition

Let $A$ be an $n \times n$ complex matrix. The matrix $A$ is normal when $$A A^* = A^* A,$$ where $A^*$ is the conjugate transpose of $A$.

Remarks

Hermitian, skew-Hermitian and unitary matrices are normal. A real matrix is normal when $A A^T = A^T A$; real symmetric and orthogonal matrices are examples. The matrix $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ is not normal. A matrix is normal if and only if its map on $\C^n$ is a normal operator for the standard inner product.

Explanations

L1 human

A matrix is normal when it commutes with its conjugate transpose. Normal matrices are exactly the matrices that a unitary change of basis makes diagonal. They include the symmetric, the Hermitian and the unitary matrices.