For a complex matrix $A$ , let $A^\ast=\overline{A}^{T}$ , where $A^T$ is the transpose, and $\bar{A}$ is the complex conjugate of $A$ .

Definition A complex square matrix $A$ is Hermitian, if $$ A = A^*. $$

Properties

1. The eigenvalues of a Hermitian matrix are real.
2. The diagonal elements of a Hermitian matrix are real.
3. The complex conjugate of a Hermitian matrix is a Hermitian matrix.
4. If $A$ is a Hermitian matrix, and $B$ is a complex matrix of same order as $A$ , then $BAB^\ast$ is a Hermitian matrix.
5. A matrix is symmetric if and only if it is real and Hermitian.
6. Hermitian matrices are a vector subspace of the vector space of complex matrices. The real symmetric matrices are a subspace of the Hermitian matrices.
7. Hermitian matrices are also called self-adjoint since if $A$ is Hermitian, then in the usual inner product of $\mathbb{C}^n$ , we have $$ \langle u,Av \rangle = \langle Au,v\rangle$$ for all $u,v\in \mathbb{C}^n$ .

Example

1. For any $n\times m$ matrix $A$ , the $n\times n$ matrix $A A^\ast$ is Hermitian.
2. For any square matrix $A$ , the Hermitian part of $A$ , $\frac{1}{2}(A+A^\ast)$ is Hermitian. See this page.
3. $$ \begin{bmatrix} 1 & 1 + i & 1 + 2i & 1 + 3i \\ 1 - i & 2 & 2 + 2i & 2 + 3i \\ 1 - 2i & 2 - 2i & 3 & 3 + 3i \\ 1 - 3i & 2 - 3i & 3 - 3i & 4 \end{bmatrix} $$

Notes

1. Hermitian matrices are named after Charles Hermite (1822-1901) [2], who proved in 1855 that the eigenvalues of these matrices are always real [1].
2. Hermitian, or self-adjoint operators on a Hilbert space play a fundamental role in quantum theories as their eigenvalues are observable, or measurable; such Hermitian operators can be represented by Hermitian matrices.

Bibliography

1

H. Eves, Elementary Matrix Theory, Dover publications, 1980.

2

The MacTutor History of Mathematics archive, Charles Hermite