direct sum of Hermitian and skewHermitian 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 skewHermitian matrix. A fancy way to say this is that complex square matrices is the direct sum of Hermitian and skewHermitian matrices.
Let us denote the vector space^{} (over $\u2102$) of complex square $n\times n$ matrices by $M$. Further, we denote by ${M}_{+}$ respectively ${M}_{}$ the vector subspaces of Hermitian and skewHermitian 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
${A}_{+}$  $=$  $\frac{1}{2}}\left(A+{A}^{\ast}\right),$  
${A}_{}$  $=$  $\frac{1}{2}}\left(A{A}^{\ast}\right).$ 
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 antiHermitian. 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

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In the special case of $1\times 1$ matrices, we obtain the decomposition of a complex number^{} into its real and imaginary components^{}.

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In the special case of real matrices, we obtain the decomposition of a $n\times n$ matrix into a symmetric matrix^{} and antisymmetric matrix.
Title  direct sum of Hermitian and skewHermitian matrices 

Canonical name  DirectSumOfHermitianAndSkewHermitianMatrices 
Date of creation  20130322 13:36:30 
Last modified on  20130322 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 