Hermitian Matrix

Definition

${\mathbf{A}}^{†}=\mathbf{A}$ $a_{ij}={\bar{a}}_{ji}$

A complex square matrix that is equal to its own Conjugate Transpose

Facts

$({\mathbf{A}}^{†}{)}^{†}=\mathbf{A}$ $(\mathbf{A}\pm \mathbf{B}{)}^{†}={\mathbf{A}}^{†}\pm {\mathbf{B}}^{†}$ $(k\mathbf{A}{)}^{†}=\overline{k}{\mathbf{A}}^{†}$ $(\mathbf{AB}{)}^{†}={\mathbf{B}}^{†}{\mathbf{A}}^{†}$

The diagonal elements of the hermitian matrix should be real numbers.

Let ${\mathbf{A}}^{†}=\mathbf{A}$ be hermitian matrix Then, ${\mathbf{x}}^{†}\mathbf{Ax}\in \mathbb{R}$

Proof $({\mathbf{x}}^{†}\mathbf{Ax}{)}^{†}={\mathbf{x}}^{†}{\mathbf{A}}^{†}\mathbf{x}={\mathbf{x}}^{†}\mathbf{Ax}$