# Hermitian matrices

A **Hermitian matrix** (or self-adjoint matrix) is a square matrix with complex elements which is equal to its own conjugate transpose — that is, the element in the th row and th column is equal to the complex conjugate of the element in the th row and th column, for all indices i and j:

If the conjugate transpose of a matrix is denoted by , then this can concisely be written as

For example,

All eigenvalues of a Hermitian matrix are real, and, moreover, eigenvectors with distinct eigenvalues are orthogonal. The typical example of a Hermitian matrix in physics is the Hamiltonian (specially in quantum mechanics).