If the conjugate transpose of a matrix A is denoted by , then this can concisely be written as
is a Hermitian matrix.
Every Hermitian matrix is normal, and the finite-dimensional spectral theorem applies. It says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This means that all eigenvalues of a Hermitian matrix are real, and, moreover, eigenvectors with distinct eigenvalues are orthogonal. It is possible to find an orthonormal basis of Cn consisting only of eigenvectors.
The sum of any two Hermitian matrices is Hermitian, and the inverse of an invertible Hermitian matrix is Hermitian as well. However, the product of two Hermitian matrices A and B will only be Hermitian if they commute, i.e., if AB = BA. Thus An is Hermitian if A is Hermitian and n is a positive integer.
The Hermitian n-by-n matrices form a vector space over the real numbers (but not over the complex numbers). The dimension of this space is n2 (one degree of freedom per main diagonal element, and two degrees of freedom per element above the main diagonal).
The eigenvectors of an Hermitian matrix are orthogonal, i.e. its eigendecomposition is where . Since right- and left- inverse are the same, we also have , and therefore (where are the eigenvalues and the eigenvectors.
Additional properties of Hermitian matrices include:
Convergence analysis of an iterative method for the reconstruction of multi-band signals from their uniform and periodic nonuniform samples.(Report)
May 01, 2008; Abstract One of the proposed methods for recovery of a band-limited signal from its samples, whether uniform or nonuniform, is...