A straightforward proof of the polynomial factorization of a positive semi-definite polynomial matrix

In this paper we present a straightforward proof of the well-known fact that a positive semi-definite polynomial matrix can be written as the product of a polynomial matrix and its associated (Hermitian) transposed. For polynomial matrices with complex coefficients both factors also have complex coefficients and can be taken of the same size as the original matrix. For polynomial matrices with real coefficients both factors may have real coefficients, but then the left factor may need twice as many columns as the original matrix, and analogously for the right factor.