Free convex sets defined by rational expressions have LMI representations

TL;DR

This paper extends the characterization of convex free sets defined by polynomial matrix inequalities to those defined by rational functions, showing they admit LMI representations, and links minimal realizations to singularity structure.

Contribution

It generalizes previous polynomial-based results to rational functions and connects minimal realizations with the singularities of free rational functions.

Findings

Convex free sets defined by rational expressions have LMI representations.

Minimal symmetric descriptor realizations encode the singularities of free rational functions.

The extension from polynomial to rational functions broadens the scope of LMI representability.

Abstract

Suppose p is a symmetric matrix whose entries are polynomials in freely noncommutating variables and p(0) is positive definite. Let D(p) denote the component of zero of the set of those g-tuples X of symmetric matrices (of the same size) such that p(X) is positive definite. By a previous result of the authors, if D(p) is convex and bounded, then D(p) can be described as the set of all solutions to a linear matrix inequality (LMI). This article extends that result from matrices of polynomials to matrices of rational functions in free variables. As a refinement of a theorem of Kaliuzhnyi-Verbovetskyi and Vinnikov, it is also shown that a minimal symmetric descriptor realization r for a symmetric free matrix-valued rational function R in g freely noncommuting variables precisely encodes the singularities of the rational function. This singularities result is an important ingredient in the proof of the LMI representation theorem stated above.