Hermitian matrix

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

a_{ij} = \overline{a_{ji}}

A = \overline {A^\text{T}}

, in matrix form.

Hermitian matrices can be understood as the complex extension of real symmetric matrices.

If the conjugate transpose of a matrix $A$ is denoted by $A^\dagger$ , then the Hermitian property can be written concisely as

A = A^\dagger.

Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues.

Examples

See the following example:

\begin{bmatrix} 2 & 2+i & 4 \\ 2-i & 3 & i \\ 4 & -i & 1 \\ \end{bmatrix}

The diagonal elements must be real, as they must be their own complex conjugate.

Well-known families of Pauli matrices, Gell-Mann matrices and their generalizations are Hermitian. In theoretical physics such Hermitian matrices are often multiplied by imaginary coefficients,^[1]^[2] which results in skew-Hermitian matrices (see below).

Here we offer another useful Hermitian matrix using an abstract example. If a square matrix $A$ equals the multiplication of a matrix and its conjugate transpose, that is, $A=BB^\dagger$ , then $A$ is a Hermitian positive semi-definite matrix. Furthermore, if $B$ is row full-rank, then $A$ is positive definite.

Properties

The entries on the main diagonal (top left to bottom right) of any Hermitian matrix are necessarily real, because they have to be equal to their complex conjugate.
Because of conjugation, for complex valued entries the off diagonal entries cannot be symmetric (or same). Hence, 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 a normal matrix.

The finite-dimensional spectral theorem says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries. This implies that all eigenvalues of a Hermitian matrix $A$ are real, and that $A$ has $n$ linearly independent eigenvectors. Moreover, it is possible to find an orthonormal basis of $C n$ consisting of $n$ eigenvectors of $A$ .

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 $B$ is Hermitian if and only if $AB = BA$ . Thus $A n$ is Hermitian if $A$ is Hermitian and $n$ is an integer.

For an arbitrary complex valued vector $v$ the product $v^\dagger A v$ is real because of $v^\dagger A v = (v^\dagger A v)^\dagger$ . This is especially important in quantum physics where hermitian matrices are operators that measure properties of a system e.g. total spin which have to be real.

The Hermitian complex $n$ -by- $n$ matrices do not form a vector space over the complex numbers, since the identity matrix $I n$ is Hermitian, but $i I n$ is not. However the complex Hermitian matrices do form a vector space over the real numbers $R$ . In the $2 n 2$ -dimensional vector space of complex $n \times n$ matrices over $R$ , the complex Hermitian matrices form a subspace of dimension $n 2$ . If $E jk$ denotes the $n$ -by- $n$ matrix with a 1 in the $j, k$ position and zeros elsewhere, a basis can be described as follows:

\; E_{jj}

for

1\leq j\leq n

(

n

matrices)

together with the set of matrices of the form

\; E_{jk}+E_{kj}

for

1\leq j<k\leq n

(n² − n/2 matrices)

and the matrices

\; i(E_{jk}-E_{kj})

for

1\leq j<k\leq n

(n² − n/2 matrices)

where

i

denotes the complex number

\sqrt{-1}

, known as the imaginary unit.

If $n$ orthonormal eigenvectors $u_1,\dots,u_n$ of a Hermitian matrix are chosen and written as the columns of the matrix $U$ , then one eigendecomposition of $A$ is $A = U \Lambda U^\dagger$ where $U U^\dagger = I=U^\dagger U$ and therefore

A = \sum _j \lambda_j u_j u_j ^\dagger

where

\lambda_j

are the eigenvalues on the diagonal of the diagonal matrix

\; \Lambda

Further properties

Additional facts related to Hermitian matrices include:

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