Solution
= Solution
The <symmetric and antisymmetric parts of a matrix> are
$$
\boxed{S=\frac12(M+M^T),\qquad A=\frac12(M-M^T)}.
$$
Indeed $S+A=M$, while the <transpose> identities $(M^T)^T=M$ give $S^T=S$ and $A^T=-A$. Thus $S$ is a <symmetric matrix> and $A$ is an <antisymmetric matrix>. The decomposition is also unique: a <matrix> that is both symmetric and antisymmetric must equal its own negative and hence vanish.