Question

Show any square matrix can be expressed as the sum of two matrices, one symmetric and the other anti-symmetric.

Answer 1

Let $A$ be the given matrix.
Then $A=\frac{1}{2}(A+A^{\prime })+\frac{1}{2}(A-A^{\prime })$
Now $(A+A^{\prime }{)}^{\prime }=A^{\prime }+(A^{\prime }{)}^{\prime }=A^{\prime }+A$
$\therefore A+A^{\prime }$ is symmetric.
or $\frac{1}{2}(A+A^{\prime })$ is symmetric.
Also, $(A-A^{\prime }{)}^{\prime }=A^{\prime }-(A^{\prime }{)}^{\prime }=A^{\prime }-A=-(A-A^{\prime })$
$\therefore A-A^{\prime }$ is anti-symmetric.
or $\frac{1}{2}(A-A^{\prime })$ is anti-symmetric.
$\therefore A=\frac{1}{2}(A+A^{\prime })+\frac{1}{2}(A-A^{\prime })=$ symmetric matrix $+$ anti-symmetric matrix