Hermitian matrix
For a complex matrix $A$, let ${A}^{\ast}={\overline{A}}^{T}$, where ${A}^{T}$ is the transpose^{}, and $\overline{A}$ 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 $BA{B}^{\ast}$ 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 selfadjoint since if $A$ is Hermitian, then in the usual inner product^{} of ${\u2102}^{n}$, we have
$$\u27e8u,Av\u27e9=\u27e8Au,v\u27e9$$ for all $u,v\in {\u2102}^{n}$.
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 (http://planetmath.org/DirectSumOfHermitianAndSkewHermitianMatrices).

3.
$$\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 1+i\hfill & \hfill 1+2i\hfill & \hfill 1+3i\hfill \\ \hfill 1i\hfill & \hfill 2\hfill & \hfill 2+2i\hfill & \hfill 2+3i\hfill \\ \hfill 12i\hfill & \hfill 22i\hfill & \hfill 3\hfill & \hfill 3+3i\hfill \\ \hfill 13i\hfill & \hfill 23i\hfill & \hfill 33i\hfill & \hfill 4\hfill \end{array}\right]$$
The first two examples are also examples of normal matrices^{}.
Notes
 1.

2.
Hermitian, or selfadjoint 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://wwwgap.dcs.stand.ac.uk/ history/Mathematicians/Hermite.htmlCharles Hermite
Title  Hermitian matrix 
Canonical name  HermitianMatrix 
Date of creation  20130322 12:12:00 
Last modified on  20130322 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  selfadjoint 
Related topic  SelfDual 
Related topic  SkewHermitianMatrix 
Related topic  SelfAdjointOperator 
Related topic  PauliMatrices 
Defines  Hermitian operator 