skew-Hermitian matrix

Definition. A square matrixMathworldPlanetmath A with complex entries is skew-Hermitian, if


Here A=AT¯, AT is the transposeMathworldPlanetmath of A, and A¯ is is the complex conjugateDlmfMathworldPlanetmath of the matrix A.


  1. 1.

    The trace of a skew-Hermitian matrix is

  2. 2.

    The eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath of a skew-Hermitian matrix are

Proof. Property (1) follows directly from property (2) since the trace is the sum of the eigenvalues. But one can also give a simple proof as follows. Let xij and yij be the real respectively imaginary partsDlmfMathworld of the elements in A. Then the diagonal elements of A are of the form xkk+iykk, and the diagonal elements in A are of the form -xkk+iykk. Hence xkk, i.e., the real part for the diagonal elements in A must vanish, and property (1) follows. For property (2), suppose A is a skew-Hermitian matrix, and x an eigenvectorMathworldPlanetmathPlanetmathPlanetmath corresponding to the eigenvalue λ, i.e.,

Ax = λx. (1)

Here, x is a complex column vectorMathworldPlanetmath. Since x is an eigenvector, x is not the zero vectorMathworldPlanetmath, and xx>0. Without loss of generality we can assume xx=1. Thus

λ¯ = xλ¯x
= (xλx)
= (xAx)
= xAx
= x(-A)x
= -xλx
= -λ.

Hence the eigenvalue λ corresponding to x is

Title skew-Hermitian matrix
Canonical name SkewHermitianMatrix
Date of creation 2013-03-22 13:36:14
Last modified on 2013-03-22 13:36:14
Owner matte (1858)
Last modified by matte (1858)
Numerical id 21
Author matte (1858)
Entry type Definition
Classification msc 15A57
Synonym anti-Hermitian matrix
Related topic HermitianMatrix
Related topic SymmetricMatrix
Related topic SkewSymmetricMatrix