  The paper proves sum-of-square-of-rational-function based representations
(shortly, sosrf-based representations) of polynomial matrices that are positive
semidefinite on some special sets: $\mathbb{R}^n;$ $\mathbb{R}$ and its
intervals $[a,b]$, $[0,\infty)$; and the strips $[a,b] \times \mathbb{R}
\subset \mathbb{R}^2.$ A method for numerically computing such representations
is also presented. The methodology is divided into two stages:
  (S1) diagonalizing the initial polynomial matrix based on the Schm\"{u}dgen's
procedure \cite{Schmudgen09};
  (S2) for each diagonal element of the resulting matrix, find its low rank
sosrf-representation satisfying the Artin's theorem solving the Hilbert's 17th
problem.
  Some numerical tests and illustrations with \textsf{OCTAVE} are also
presented for each type of polynomial matrices.
