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.

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(b)

(c)