Any square matrix can be written as a sum of matrix and its transpose
A=\[\frac{A+A'}{2}+\frac{A-A'}{2}\] where A is given matrix and \[A'\] is transpose of the given matrix.
\[A+A'\] is symmetric matrix and \[A-A'\] is a skew symmetric matrix.
Example- Express the following matrices as the sum of a symmetric and a skew symmetric matrix:
\[\begin{bmatrix}3&5\\1&-1\end{bmatrix}\]
solution-
\[\frac { \begin{bmatrix}3&5\\1&-1\end{bmatrix}+\begin{bmatrix}3&1\\5&-1\end{bmatrix}}{2}=\begin{bmatrix}3&3\\3&-1\end{bmatrix}\]..............(1)
which is a symmetric matrix
\[\frac {\begin{bmatrix}3&5\\1&-1\end{bmatrix}-\begin{bmatrix}3&1\\5&-1\end{bmatrix}}{2}\]=\[\begin{bmatrix}0&2\\-2&0\end{bmatrix}\].................(2)
which is skew symmetric matrix
adding equation 1 and 2 we get matrix A as\[\begin{bmatrix}3&5\\1&-1\end{bmatrix}\]
