Question

Example of a matrix which is hermitian and unitary

Answer 1

Commutative matrices

Two square matrices ${\bf A}$ and ${\bf B}$ commute if ${\bf A}{\bf B}={\bf B}{\bf A}$.

Obviously all diagonal matrices commute.

Hermitian matrix

A square matrix ${\bf A}$ is a Hermitian matrix if it is equal to its complex conjugate transpose ${\bf A}^*=\overline{\bf A}^T={\bf A}$. If a Hermitian matrix $\overline{\bf A}={\bf A}$ is real, it is a symmetric matrix, ${\bf A}^T={\bf A}$.

Unitary matrix

${\bf A}$ is a unitary matrix if its conjugate transpose is equal to its inverse ${\bf A}^*={\bf A}^{-1}$, i.e., ${\bf A}^* {\bf A}={\bf I}$. When a unitary matrix $\overline{\bf A}={\bf A}$ is real, it becomes an orthogonal matrix,