Hermite Matrix

Hermite Matrix

Definition

In mathematics, a Hermitian matrix(or self-joint matrix) is a complex square matrix 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: \[ a_{i j}=\overline{a_{j i}} \] or in matrix form \[ A=\overline{A^{\top}} \] Hermitian matrices can be understood as the complex extension of real symmetric matrices If a conjugate transpose of a matrix A is denoted by \(A^H\), then the Hermitian property can be written concisely as \[ A = A^{H} \] Other equivalent notations in common use are \[ A^{\mathrm{H}}=A^{\dagger}=A^{*} \]

Alternative characterizations

Equality with the adjoint

A square matrix A is Hermitian if and only if it is equal to its adjoint, that is, it satisfies \[ \left<w, Av\right> = \left<Aw, v\right> \] where any pair of vectors v, w, where <.,.> denotes the inner product operation

Reality of quadratic forms

A square matrix A is Hermitian if and only if \[ \langle\mathbf{v}, A \mathbf{v}\rangle \in \mathbb{R}, \quad \mathbf{v} \in \mathbb{R} . \]

Spectral properties

A square matrix A is Hermitian if and only if it is unitarily diagonalizable with real eigenvalues

Properties

Main diagonal values are real

Symmetric

A matrix that has only real entries is symmetric is and symmetric if and only if it is a Hermitian matrix. A real and symmetric matrix is simply a special case of a Hermitian matrix

Normal

Every Hermitian matrix is a normal matrix. \[ AA^H = A^HA \]

Diagonalizable

Sum of Hermitian matrices

\[ (A+B)_{i j}=A_{i j}+B_{i j}=\bar{A}_{j i}+\bar{B}_{j i}=\overline{(A+B)}_{j i} \]

Inverse is Hermitian

ABA Hermitian

Real determinant

The determinant of a Hermitian matrix is real