direct sum of Hermitian and skew-Hermitian matrices

In this example, we show that any square matrixMathworldPlanetmath with complex entries can uniquely be decomposed into the sum of one Hermitian matrixMathworldPlanetmath and one skew-Hermitian matrix. A fancy way to say this is that complex square matrices is the direct sum of Hermitian and skew-Hermitian matrices.

Let us denote the vector spaceMathworldPlanetmath (over ) of complex square n×n matrices by M. Further, we denote by M+ respectively M- the vector subspaces of Hermitian and skew-Hermitian matrices. We claim that

M = M+M-. (1)

Since M+ and M- are vector subspaces of M, it is clear that M++M- is a vector subspace of M. Conversely, suppose AM. We can then define

A+ = 12(A+A),
A- = 12(A-A).

Here A=A¯T, and A¯ is the complex conjugateMathworldPlanetmath of A, and AT is the transposeMathworldPlanetmath of A. It follows that A+ is Hermitian and A- is anti-Hermitian. Since A=A++A-, any element in M can be written as the sum of one element in M+ and one element in M-. Let us check that this decomposition is unique. If AM+M-, then A=A=-A, so A=0. We have established equation 1.

Special cases

Title direct sum of Hermitian and skew-Hermitian matrices
Canonical name DirectSumOfHermitianAndSkewHermitianMatrices
Date of creation 2013-03-22 13:36:30
Last modified on 2013-03-22 13:36:30
Owner mathcam (2727)
Last modified by mathcam (2727)
Numerical id 5
Author mathcam (2727)
Entry type Example
Classification msc 15A03
Classification msc 15A57
Related topic DirectSumOfEvenoddFunctionsExample