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 $\mathbb{C}$ ) of complex square $n\times n$ matrices by $M$ . Further, we denote by $M_{+}$ respectively $M_{-}$ the vector subspaces of Hermitian and skew-Hermitian matrices. We claim that

\displaystyle M

\displaystyle=

\displaystyle M_{+}\oplus 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 $A\in M$ . We can then define

	$\displaystyle A_{+}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\big{(}A+A^{\ast}\big{)},$
	$\displaystyle A_{-}$	$\displaystyle=$	$\displaystyle\frac{1}{2}\big{(}A-A^{\ast}\big{)}.$

Here $A^{\ast}=\overline{A}\hskip 1.13811pt^{\mbox{\scriptsize{T}}}\hskip 0.569055pt$ , and $\overline{A}$ is the complex conjugate of $A$ , and $A\hskip 1.13811pt^{\mbox{\scriptsize{T}}}\hskip 0.569055pt$ is the transpose 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 $A\in M_{+}\cap M_{-}$ , then $A=A^{\ast}=-A$ , so $A=0$ . We have established equation 1.

Special cases

•

In the special case of $1\times 1$ 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 $n\times n$ 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