Hermitian matrix

A Hermitian matrix is one that satisfies <math>M=M^\dagger</math>, where <math>M^\dagger</math> is the Hermitian conjugate of <math>M</math> (i.e., the matrix formed by transposing <math>M</math> and taking the complex conjugate of each element). As an example, the most general 2x2 Hermitian matrix has the form

<math> \begin{pmatrix}

 a      & b      \\
 b^*      & c 

\end{pmatrix} </math>

for any complex number <math>b</math> and any real numbers <math>a</math> and <math>c</math>. In the case where all elements of the matrix are real, a Hermitian matrix becomes symmetric (as Hermitian conjugation then becomes equivalent to transposition).

Properties of Hermitian matrices

• The eigenvalues are all real.
• The eigenvectors corresponding to different eigenvalues are orthogonal.

Because of these properties, Hermitian matrices have important applications in quantum mechanics.