Editing Hermitian matrix (section)

==Alternative characterizations==

Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:

===Equality with the adjoint===

A square matrix <math>A</math> is Hermitian if and only if it is equal to its [[Hermitian adjoint|conjugate transpose]], that is, it satisfies
<math display="block">\langle \mathbf w, A \mathbf v\rangle = \langle A \mathbf w, \mathbf v\rangle,</math>
for any pair of vectors <math>\mathbf v, \mathbf w,</math> where <math>\langle \cdot, \cdot\rangle</math> denotes [[Dot product|the inner product]] operation.

This is also the way that the more general concept of [[self-adjoint operator]] is defined.

===Real-valuedness of quadratic forms===

An <math>n\times{}n</math> matrix <math>A</math> is Hermitian if and only if
<math display="block">\langle \mathbf{v}, A \mathbf{v}\rangle\in\R, \quad \text{for all } \mathbf{v}\in \mathbb{C}^{n}.</math>

===Spectral properties===

A square matrix <math>A</math> is Hermitian if and only if it is unitarily [[Diagonalizable matrix|diagonalizable]] with real [[Eigenvalues and eigenvectors|eigenvalues]].