Bilinear matrix inequalities and polynomials in several freely noncommuting variables

TL;DR

This paper characterizes matrix-valued polynomials with specific convexity properties in noncommuting variables using algebraic certificates involving Linear Matrix Inequalities and Bilinear Matrix Inequalities.

Contribution

It provides a novel algebraic characterization of certain noncommutative polynomials through LMIs and BMIs, expanding the understanding of convexity in free variables.

Findings

Characterization of matrix-valued polynomials with partial convexity

Use of algebraic certificates involving LMIs and BMIs

Extension of convexity concepts to noncommuting variables

Abstract

Matrix-valued polynomials in any finite number of freely noncommuting variables that enjoy certain canonical partial convexity properties are characterized, via an algebraic certificate, in terms of Linear Matrix Inequalities and Bilinear Matrix Inequalities.