2012/09/15 by J. William Helton, Scott McCullough, Helton, J. William +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #math.FA
paper · pdf · doi:10.48550/arxiv.1209.3412
arxiv created 2012/11/21 · arxiv updated 2012/11/22
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.