# Hermitian matrix

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. 1.

The eigenvalues of a Hermitian matrix are real.

2. 2.

The diagonal elements of a Hermitian matrix are real.

3. 3.

The complex conjugate of a Hermitian matrix is a Hermitian matrix.

4. 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. 5.

A matrix is symmetric if and only if it is real and Hermitian.

6. 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. 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. 1.

For any $n\times m$ matrix $A$, the $n\times n$ matrix $AA^{\ast}$ is Hermitian.

2. 2.

For any square matrix $A$, the Hermitian part of $A$, $\frac{1}{2}(A+A^{\ast})$ is Hermitian. See this page (http://planetmath.org/DirectSumOfHermitianAndSkewHermitianMatrices).

3. 3.
 $\begin{bmatrix}1&1+i&1+2i&1+3i\\ 1-i&2&2+2i&2+3i\\ 1-2i&2-2i&3&3+3i\\ 1-3i&2-3i&3-3i&4\end{bmatrix}$

The first two examples are also examples of normal matrices.

## Notes

1. 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. 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.

## References

• 1 H. Eves, Elementary Matrix Theory, Dover publications, 1980.
• 2 The MacTutor History of Mathematics archive, http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Hermite.htmlCharles Hermite
 Title Hermitian matrix Canonical name HermitianMatrix Date of creation 2013-03-22 12:12:00 Last modified on 2013-03-22 12:12:00 Owner matte (1858) Last modified by matte (1858) Numerical id 21 Author matte (1858) Entry type Definition Classification msc 15A57 Synonym Hermitian Synonym self-adjoint Related topic SelfDual Related topic SkewHermitianMatrix Related topic SelfAdjointOperator Related topic PauliMatrices Defines Hermitian operator