  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 $M=Q^TQ$, 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 $M=Q^TQ$ 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
Fej\'er-Riesz Theorem.
