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

## TL;DR

This paper establishes sum-of-square-of-rational-function representations for positive semidefinite polynomial matrices on various sets, providing a numerical method and demonstrating its effectiveness through tests.

## Contribution

It introduces a novel method for representing positive semidefinite polynomial matrices using sosrf-based forms on specific sets, with a two-stage diagonalization and low-rank approximation approach.

## Key findings

- Method successfully computes sosrf-representations for various polynomial matrices.
- Numerical tests validate the approach's accuracy and efficiency.
- Applicable to matrices positive on real line, intervals, and strips.

## Abstract

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.