## Hermitian Matrix

If a Matrix is Self-Adjoint, it is said to be a Hermitian matrix. Therefore, a Hermitian Matrix is defined as one for which

 (1)

where denotes the Adjoint Matrix. Hermitian Matrices have Real Eigenvalues with Orthogonal Eigenvectors. For Real Matrices, Hermitian is the same as symmetrical. Any Matrix which is not Hermitian can be expressed as the sum of two Hermitian matrices
 (2)

Let be a Unitary Matrix and be a Hermitian matrix. Then the Adjoint Matrix of a Similarity Transformation is
 (3)

The specific matrix

 (4)

where are Pauli Spin Matrices, is sometimes called the'' Hermitian matrix.

Arfken, G. Hermitian Matrices, Unitary Matrices.'' §4.5 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 209-217, 1985.