Let $m$ and $n$ be two positive integers. For the free group $F_n$ generated by $n$ letters, and a symmetric polynomial $b$ with variables in $F_n$ and with $n$-by-$n$ complex matrices coeffitients, we use real algebraic geometry to give a new proof showing that $b$ is a sum of Hermitian squares if and only if $b$ is mapped to a positive semidefintie matrix under any finitely dimensional unitary representation of $F_n$.