# 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 Template:Mvar-th row and Template:Mvar-th column is equal to the complex conjugate of the element in the Template:Mvar-th row and Template:Mvar-th column, for all indices Template:Mvar and Template:Mvar:

${\displaystyle a_{ij}={\overline {a_{ji}}}}$ or ${\displaystyle A={\overline {A^{\text{T}}}}}$, in matrix form.

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

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

${\displaystyle A=A^{\dagger }.}$

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

## Examples

See the following example:

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

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 ${\displaystyle A}$ equals the multiplication of a matrix and its conjugate transpose, that is, ${\displaystyle A=BB^{\dagger }}$, then ${\displaystyle A}$ is a Hermitian positive semi-definite matrix. Furthermore, if ${\displaystyle B}$ is row full-rank, then ${\displaystyle A}$ 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. 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.
• The Hermitian complex Template:Mvar-by-Template:Mvar matrices do not form a vector space over the complex numbers, since the identity matrix In is Hermitian, but iIn is not. However the complex Hermitian matrices do form a vector space over the real numbers R. In the 2n2-dimensional vector space of complex n × n matrices over R, the complex Hermitian matrices form a subspace of dimension n2. If Ejk denotes the Template:Mvar-by-Template:Mvar matrix with a 1 in the j,k position and zeros elsewhere, a basis can be described as follows:
${\displaystyle \;E_{jj}}$ for ${\displaystyle 1\leq j\leq n}$ (Template:Mvar matrices)
together with the set of matrices of the form
${\displaystyle \;E_{jk}+E_{kj}}$ for ${\displaystyle 1\leq j ({{ safesubst:#invoke:Unsubst||$B=n2n/2}} matrices) and the matrices ${\displaystyle \;i(E_{jk}-E_{kj})}$ for ${\displaystyle 1\leq j ({{ safesubst:#invoke:Unsubst||$B=n2n/2}} matrices)
where ${\displaystyle i}$ denotes the complex number ${\displaystyle {\sqrt {-1}}}$, known as the imaginary unit.
${\displaystyle A=\sum _{j}\lambda _{j}u_{j}u_{j}^{\dagger }}$,
where ${\displaystyle \lambda _{j}}$ are the eigenvalues on the diagonal of the diagonal matrix ${\displaystyle \;\Lambda }$.

## Further properties

{{safesubst:#invoke:anchor|main}}Additional facts related to Hermitian matrices include:

${\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}(C+C^{\dagger })\quad {\mbox{and}}\quad B={\frac {1}{2}}(C-C^{\dagger }).}$
• The determinant of a Hermitian matrix is real:
Proof: ${\displaystyle \det(A)=\det(A^{\mathrm {T} })\quad \Rightarrow \quad \det(A^{\dagger })=\det(A)^{*}}$
Therefore if ${\displaystyle A=A^{\dagger }\quad \Rightarrow \quad \det(A)=\det(A)^{*}.}$
(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)

## Rayleigh quotient

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