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 $ℂ$) 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

$M$ $=$ $M_{+}\oplus M_{-}.$ (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

$A_{+}$ $=$ ${\displaystyle \frac{1}{2}}\left(A+A^{\ast }\right),$

$A_{-}$ $=$ ${\displaystyle \frac{1}{2}}\left(A-A^{\ast }\right).$

Here $A^{\ast }={\overline{A}}^{\text{T}}$, and $\overline{A}$ is the complex conjugate of $A$, and $A^{\text{T}}$ 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