Definitions

Hermitian matrix

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

$a_\left\{i,j\right\} = overline\left\{a_\left\{j,i\right\}\right\}.$

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

$A = A^*. ,$

For example,

$begin\left\{bmatrix\right\}3&2+i$
2-i&1end{bmatrix}

is a Hermitian matrix.

Properties

The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are necessarily real. A matrix that has only real entries is Hermitian if and only if it is a symmetric matrix, i.e., if it is symmetric with respect to the main diagonal. A real and symmetric matrix is simply a special case of 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 $A = U Sigma U^dagger$ where $U U^dagger = I$. Since right- and left- inverse are the same, we also have $U^dagger U = I$, and therefore $A = sum _i sigma_i u_i u_i ^dagger$ (where $sigma_i$ are the eigenvalues and $u_i$ the eigenvectors.

If the eigenvalues of a Hermitian matrix are all positive, then the matrix is positive definite; if they are all non-negative, then the matrix is positive semidefinite.

Additional properties of Hermitian matrices include:

• The sum of a square matrix and its conjugate transpose $\left(C + C^\left\{dagger\right\}\right)$ is Hermitian.
• The difference of a square matrix and its conjugate transpose $\left(C - C^\left\{dagger\right\}\right)$ is skew-Hermitian (also called antihermitian).
• An arbitrary square matrix C can be written as the sum of a Hermitian matrix A and a skew-Hermitian matrix B:

$C = A+B quadmbox\left\{with\right\}quad A = frac\left\{1\right\}\left\{2\right\}\left(C + C^\left\{dagger\right\}\right) quadmbox\left\{and\right\}quad B = frac\left\{1\right\}\left\{2\right\}\left(C - C^\left\{dagger\right\}\right).$