Hermitian Matrix
Definition
Let $A$ be a square complex matrix. If $A$ satisfies the following equation, it is called a Hermitian matrix or self-adjoint matrix.
$$ A^{\ast}=A $$
Here, $A^{\ast}$ is the conjugate transpose of $A$. If $A$ satisfies the following equation, it is called a skew-Hermitian matrix .
$$ A^{\ast}=-A $$
Explanation
If it is a real matrix, since $A^{\ast}=A^{T}$, if it is a symmetric matrix, it is a Hermitian matrix. As can be seen from the following properties, the diagonal elements of a Hermitian matrix must be real. Therefore, if the matrix is small, it is easy to see at a glance whether it is a Hermitian matrix.
For the same reason that the diagonal elements of a Hermitian matrix must be real, the diagonal elements of a skew-Hermitian matrix are all $0$.
Properties
Let $A$ be a Hermitian matrix.
(a) The diagonal elements of $A$ must be real.
(b) The eigenvalues of $A$ are all real.
(c) Eigenvectors having different eigenvalues of $A$ are orthogonal to each other.
(b) In the context of quantum mechanics, ‘The expectation value of a Hermitian operator is always real’.
Proof
(a)
The transpose $A^{T}$ of matrix $A$ is obtained by reflecting the elements of $A$ across the main diagonal. Therefore, the diagonal elements of both matrices are always the same. This means $a_{ij}=\overline{a_{ij}}$, so the diagonal elements are real.
■