Question

Prove that any square matrix $A$ can be expressed as the sum of two symmetric and skew-symmetric matrices.

Answer 1

Any square matrix $A$ can be expressed as,

$A=\frac{A+A^T}{2}+\frac{A-A^T}{2}$.

or, $A=P+Q$ ( Let ).

Where,

$P=\frac{A+A^T}{2}$ and $Q=\frac{A-A^T}{2}$.

Now, we have, $P^T=P$ and $Q^T=-Q$.

So we've $P$ to be symmetric and $Q$ to be skew-symmetric matrix.

So any square matrix $A$ can be expressed as the sum of two symmetric and skew-symmetric matrices.