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Journal of Convex Analysis 21 (2014), No. 2, 425--448
Copyright Heldermann Verlag 2014

Free Convex Sets Defined by Rational Expressions Have LMI Representations

J. William Helton
Dept. of Mathematics, University of California, 9500 Gilman Drive, La Jolla, CA 92093-0112, U.S.A.

Scott McCullough
Dept. of Mathematics, University of Florida, Box 118105, Gainesville, FL 32611-8105, U.S.A.


\def\cD{{\mathcal D}} Suppose $p$ is a symmetric matrix whose entries are polynomials in freely noncommutative variables and $p(0)$ is positive definite. Let $\cD_p$ denote the component of zero of the set of those $g$-tuples $X=(X_1,\dots,X_g)$ of symmetric matrices (of the same size) such that $p(X)$ is positive definite. In another paper of the authors [{\it Every free convex basic semi-algebraic set has an LMI representation}, Annals of Mathematics, to appear] it was shown that if $\cD_p$ is convex and bounded, then $\cD_p$ can be described as the set of solutions of a linear matrix inequality (LMI). This article extends that result from matrices of polynomials to matrices of rational functions in free variables.\par 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 $\mathfrak{r}$ in $g$ freely noncommuting variables $x=(x_1,\dots,x_g)$ 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.

Keywords: Matrix convexity, free convexity, linear matrix inequality, noncommutative rational function, free rational function.

MSC: 47Axx; 47A63, 47L07, 14P10

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