Hermitian matrix

A Hermitian matrix is studied in the mathematical subfield of linear algebra. It is a special type of square matrices of coefficients in the complex numbers. The analog concept over the real numbers is the symmetric matrix. Named is the Hermitian matrix after the mathematician Charles Hermite.

Definition

A matrix with entries in is called Hermitian if it coincides with its ( Hermitian ) adjoint, ie the transposed and complex conjugated matrix. That is, when

Applies.

For the adjoint matrix to find the names and. The notation is also used for the complex conjugate matrix.

Therefore applies to the entries of a Hermitian matrix:

In other words, a matrix is Hermitian if and only if its transpose is equal to its complex conjugate, ie.

Examples

• The matrix

• The Pauli matrices

Properties

Direct consequences of the definition

Self-adjoint

Hermitian matrices are self-adjoint with respect to the Standardskalarprodukts on: For all

It follows:

• Hermitian matrices can be diagonalized always.
• Real symmetric matrices can be diagonalized real.
• The eigenvalues ​​of a Hermitian matrix are real.
• There exists an orthogonal basis of eigenvectors.

In addition:

• The determinant of a Hermitian matrix is ​​real.

• The sum of two Hermitian matrices is Hermitian.
• The product of two Hermitian matrices is Hermitian if they commute.
• The powers of Ak with k ≥ 0 and the invertibility are Hermitian with k < 0.
• An arbitrary square matrix C can be clearly identified as the sum of a Hermitian matrix A and a matrix B schiefhermiteschen be written: with and
• In the 2n2 -dimensional vector space of complex n × n- matrices, the Hermitian one - subspace of dimension n2 form. However, they do not form a subspace, because the i times the identity matrix is not Hermitian.