  Positivstellens{\"a}tze are a group of theorems on the positivity of
involution algebras over $\mathbb{R}$ or $\mathbb{C}$. One of the most
well-known Positivstellensatz is the solution to Hilbert's 17th problem given
by E. Artin, which asserts that a real polynomial in $n$ commutative variables
is nonnegative on real affine space if and only if it is a sum of fractional
squares. 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 coefficients, 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 semidefinite matrix under any finitely
dimensional unitary representation of $F_n$.
