Finsler's Lemma for Matrix Polynomials

Abstract

Finsler's Lemma charactrizes all pairs of symmetric $n\times n$ real matrices $A$ and $B$ which satisfy the property that $v^{T}Av>0$ for every nonzero $v\in {\mathbb{R}}^n$ such that $v^{T}Bv=0$. 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.

23 pages, 2 figures, submitted

References in corpus (6)

• Moment problems for operator polynomials
• Noncommutative polynomials nonnegative on a variety intersect a convex set
• On real one-sided ideals in a free algebra
• A Real Nullstellensatz for Free Modules
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Cited by in corpus (1)

• Existence of Continuous or Constant Finsler's Variables for Parameter-Dependent Systems