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.
Categories: math.OC cs.CL