[en] We study sum-of-squares representations of symmetric univariate real matrix polynomials that are positive semidefinite along the real line. We give a new proof of the fact that every positive semidefinite univariate matrix polynomial of size $n\times <!--\; ??\; -->n$ can be written as a sum of squares $M=Q^{T}Q$, where Q has size $(n+1)\times <!--\; ??\; -->n$, which was recently proved by Blekherman–Plaumann–Sinn–Vinzant. Our new approach using the theory of quadratic forms allows us to prove the conjecture made by these authors that these minimal representations $M=Q^{T}Q$ are generically in one-to-one correspondence with the representations of the nonnegative univariate polynomial $det(M)$ as sums of two squares. In parallel, we will use our methods to prove the more elementary hermitian analogue that every hermitian univariate matrix polynomial M that is positive semidefinite along the real line, is a square, which is known as the matrix Fejér–Riesz Theorem.