Hermitian matrix

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

${\begin{pmatrix}a&b\\b^{*}&c\end{pmatrix}}$

for arbitrary complex numbers $a,b,c$. 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.