# Hermitian matrix

Hermitian-symmetric matrix, self-conjugate matrix

A square matrix $A = \| a _ {ik} \|$ over $\mathbf C$ that is the same as its Hermitian-conjugate matrix

$$A ^ {*} = \overline{A}\; {} ^ {T} = \| \overline{ {a _ {ki} }}\; \| ,$$

that is, a matrix whose entries satisfy the condition $a _ {ik} = \overline{ {a _ {ki} }}\;$. If all the $a _ {ik} \in \mathbf R$, then a Hermitian matrix is symmetric (cf. Symmetric matrix). The Hermitian matrices of a fixed order form a vector space over $\mathbf R$. If $A$ and $B$ are two Hermitian matrices of the same order, then so is $AB + BA$. Under the operation $A \cdot B = ( AB + BA ) / 2$ the Hermitian matrices (of order $n$) form a Jordan algebra. The product $AB$ of two Hermitian matrices is itself Hermitian if and only if $A$ and $B$ commute.

The Hermitian matrices of order $n$ are the matrices of Hermitian transformations of an $n$- dimensional unitary space in an orthonormal basis (see Self-adjoint linear transformation). On the other hand, Hermitian matrices are the matrices of Hermitian forms in an $n$- dimensional complex vector space. Like Hermitian forms (cf. Hermitian form), Hermitian matrices can be defined over any skew-field with an anti-involution.

All eigen values of a Hermitian matrix are real. For every Hermitian matrix $A$ there exists a unitary matrix $U$ such that $U ^ {-} 1 AU$ is a real diagonal matrix. A Hermitian matrix is called non-negative (or positive semi-definite) if all its principal minors are non-negative, and positive definite if they are all positive. Non-negative (positive-definite) Hermitian matrices correspond to non-negative (positive-definite) Hermitian linear transformations and Hermitian forms.