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 $A$.

DefinitionA 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 matrixof 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 ofcomplex matrices.The real symmetric matrices are a subspace of the Hermitian matrices.

7. Hermitian matrices are also called self-adjoint since if $A$ isHermitian, then in the usualinner product of ${ℂ}^n$, we have

   $$⟨u,Av⟩=⟨Au,v⟩$$

   for all $u,v\in {ℂ}^n$.

Example

1. For any $n\times m$ matrix $A$, the $n\times n$ matrix $A{A}^{\ast }$ isHermitian.

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).

The first two examples are also examples of normal matrices.

Notes

1. Hermitian matrices are named after Charles Hermite (1822-1901) [2], who proved in 1855 that theeigenvalues of these matrices are always real [1].

2. Hermitian, or self-adjoint operators on a Hilbert space play a fundamentalrole in quantum theories as their eigenvalues are observable, or measurable; suchHermitian operators can be represented by Hermitian matrices.