Question

Express the following matrix as the sum of a symmetric and a skew-symmetric matrices;

$\left[\begin{array}{cc}1& 5\\ -1& 2\end{array}\right]$

Answer 1

Let $A=\left[\begin{array}{cc}1& 5\\ -1& 2\end{array}\right]$. Then, $A^{\prime }={\left[\begin{array}{cc}1& 5\\ -1& 2\end{array}\right]}^{\prime }=\left[\begin{array}{cc}1& -1\\ 5& 2\end{array}\right]$
Now, $A+A^{\prime }=\left[\begin{array}{cc}1& 5\\ 4& 4\end{array}\right]+\left[\begin{array}{cc}1& -1\\ 5& 2\end{array}\right]=\left[\begin{array}{cc}2& 4\\ 4& 4\end{array}\right]$

Let $P=\frac{1}{2}(A+A^{\prime })=\frac{1}{2}\left[\begin{array}{cc}2& 4\\ 4& 4\end{array}\right]=\left[\begin{array}{cc}1& 2\\ 2& 2\end{array}\right],Now,P^{\prime }=\left[\begin{array}{cc}1& 2\\ 2& 2\end{array}\right]=\left[\begin{array}{cc}1& 2\\ 2& 2\end{array}\right]=P$
Thus, $P=\frac{1}{2}(A+A^{\prime })$ is a symmetric matrix.
Now, $A-A^{\prime }=\left[\begin{array}{cc}1& 5\\ -1& 2\end{array}\right]-\left[\begin{array}{cc}1& -1\\ 5& 2\end{array}\right]=\left[\begin{array}{cc}0& 6\\ -6& 0\end{array}\right]$
Let $Q=\frac{1}{2}(A-A^{\prime })=\frac{1}{2}\left[\begin{array}{cc}0& 6\\ -6& 0\end{array}\right]=\left[\begin{array}{cc}0& 3\\ -3& 0\end{array}\right]$
Now, $Q^{\prime }={\left[\begin{array}{cc}0& 3\\ -3& 0\end{array}\right]}^{\prime }=\left[\begin{array}{cc}0& -3\\ 3& 0\end{array}\right]=-Q$
Thus, $Q=\frac{1}{2}(A-A^{\prime })$ is a skew -symmetric matrix.
Represending A as the sum of P and Q.
$P+Q=\left[\begin{array}{cc}1& 2\\ 2& 2\end{array}\right]+\left[\begin{array}{cc}0& 3\\ -3& 0\end{array}\right]=\left[\begin{array}{cc}1& 5\\ -1& 2\end{array}\right]=A$