%0 Journal Article
%T Finsler's Lemma for Matrix Polynomials
%A Jaka Cimpric
%J Mathematics
%D 2014
%I arXiv
%R 10.1016/j.laa.2014.09.037
%X Finsler's Lemma charactrizes all pairs of symmetric $n \times n$ real matrices $A$ and $B$ which satisfy the property that $v^T A v>0$ for every nonzero $v \in \mathbb{R}^n$ such that $v^T B v=0$. We extend this characterization to all symmetric matrices of real multivariate polynomials, but we need an additional assumption that $B$ is negative semidefinite outside some ball. We also give two applications of this result to Noncommutative Real Algebraic Geometry which for $n=1$ reduce to the usual characterizations of positive polynomials on varieties and on compact sets.
%U http://arxiv.org/abs/1406.7442v1