direct sum of Hermitian and skew-Hermitian matrices
In this example, we show that any square matrix with complex
entries can uniquely be decomposed into the sum of one Hermitian matrix
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 space (over ) of
complex square matrices by .
Further, we denote by respectively the vector
subspaces of Hermitian and skew-Hermitian matrices.
We claim that
(1) |
Since and are vector subspaces of , it is clear that is a vector subspace of . Conversely, suppose . We can then define
Here , and is the complex conjugate of ,
and is the transpose
of . It follows that is Hermitian
and is anti-Hermitian. Since , any element
in can be written as
the sum of one element in and one element in . Let us check
that this decomposition is unique. If , then
, so .
We have established equation 1.
Special cases
-
•
In the special case of matrices, we obtain the decomposition of a complex number
into its real and imaginary components
.
-
•
In the special case of real matrices, we obtain the decomposition of a matrix into a symmetric matrix
and anti-symmetric matrix.
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 |