direct sum of Hermitian and skew-Hermitian matricesDirectSumOfHermitianAndSkewHermitianMatrices2003-05-03 05:48:062006-06-20 13:14:11ExampleHermitian matrix% this is the default PlanetMath preamble. as your knowledge
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\def\dtra{\hspace{0.04cm} ^{\mbox{\scriptsize{T}}} \hspace{0.02cm}}
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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 $\matC$.
Further, we denote by $\matCp$ respectively $\matCm$ the vector
subspaces of Hermitian and skew-Hermitian matrices.
We claim that
\begin{eqnarray}
\label{eqp}
\matC &=& \matCp \oplus \matCm.
\end{eqnarray}
Since $\matCp$ and $\matCm$ are vector subspaces of $\matC$, it is clear
that $\matCp +\matCm$ is a vector subspace of $\matC$. Conversely, suppose
$A\in \matC$. We can then define
\begin{eqnarray*}
A_+ &=& \frac{1}{2}\big( A + A^\ast \big), \\
A_- &=& \frac{1}{2}\big( A - A^\ast \big).
\end{eqnarray*}
Here $A^\ast = \ccj{A}\dtra$, and $\ccj{A}$ is the complex conjugate of $A$,
and $A\dtra$ is the transpose of $A$. It follows that $A_+$ is Hermitian
and $A_-$ is anti-Hermitian. Since $A=A_+ + A_-$, any element
in $\matC$ can be written as
the sum of one element in $\matCp$ and one element in $\matCm$. Let us check
that this decomposition is unique. If $A\in \matCp\cap \matCm$, then
$A=A^\ast=-A$, so $A=0$.
We have established equation \ref{eqp}.
{\bf Special cases}
\begin{itemize}
\item In the special case of $1\times 1$ matrices, we obtain the
decomposition of a complex number into its real and imaginary components.
\item 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.
\end{itemize}