Sum-of-square-of-rational-function based representations of positive semidefinite polynomial matrices

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.