Hermitian matrix

For a complex matrix A, let A=A¯T, where AT is the transposeMathworldPlanetmath, and A¯ is the complex conjugateMathworldPlanetmath of A.

Definition A complex square matrixMathworldPlanetmath A is Hermitian, if



  1. 1.

    The eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of a Hermitian matrix are real.

  2. 2.

    The diagonal elements of a Hermitian matrix are real.

  3. 3.

    The complex conjugate of a Hermitian matrix is a Hermitian matrix.

  4. 4.

    If A is a Hermitian matrix, and B is a complex matrix of same order as A, then BAB is a Hermitian matrix.

  5. 5.

    A matrix is symmetricPlanetmathPlanetmath if and only if it is real and Hermitian.

  6. 6.

    Hermitian matrices are a vector subspace of the vector spaceMathworldPlanetmath of complex matrices. The real symmetric matrices are a subspacePlanetmathPlanetmath of the Hermitian matrices.

  7. 7.

    Hermitian matrices are also called self-adjoint since if A is Hermitian, then in the usual inner productMathworldPlanetmath of n, we have


    for all u,vn.


  1. 1.

    For any n×m matrix A, the n×n matrix AA is Hermitian.

  2. 2.

    For any square matrix A, the Hermitian part of A, 12(A+A) is Hermitian. See this page (http://planetmath.org/DirectSumOfHermitianAndSkewHermitianMatrices).

  3. 3.

The first two examples are also examples of normal matricesMathworldPlanetmath.


  1. 1.

    Hermitian matrices are named after Charles Hermite (1822-1901) [2], who proved in 1855 that the eigenvalues of these matrices are always real [1].

  2. 2.

    Hermitian, or self-adjoint operators on a Hilbert spaceMathworldPlanetmath play a fundamental role in quantum theories as their eigenvalues are observable, or measurable; such Hermitian operators can be represented by Hermitian matrices.


  • 1 H. Eves, Elementary MatrixMathworldPlanetmath Theory, Dover publications, 1980.
  • 2 The MacTutor History of Mathematics archive, http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Hermite.htmlCharles Hermite
Title Hermitian matrix
Canonical name HermitianMatrix
Date of creation 2013-03-22 12:12:00
Last modified on 2013-03-22 12:12:00
Owner matte (1858)
Last modified by matte (1858)
Numerical id 21
Author matte (1858)
Entry type Definition
Classification msc 15A57
Synonym Hermitian
Synonym self-adjoint
Related topic SelfDual
Related topic SkewHermitianMatrix
Related topic SelfAdjointOperator
Related topic PauliMatrices
Defines Hermitian operator