Abstract

Suppose pp is a symmetric matrix whose entries are polynomials in freely noncommutative variables and p(0)p(0) is positive definite. Let Dp\cD_p denote the component of zero of the set of those gg-tuples X=(X1,…,Xg)X=(X_1,\dots,X_g) of symmetric matrices (of the same size) such that p(X)p(X) is positive definite. In another paper of the authors [Every free convex basic semi-algebraic set has an LMI representation, Annals of Mathematics, to appear] it was shown that if Dp\cD_p is convex and bounded, then Dp\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. As a refinement of a theorem of Kaliuzhnyi-Verbovetskyi and Vinnikov, it is also shown that a minimal symmetric descriptor realization rr for a symmetric free matrix-valued rational function r\mathfrak{r} in gg freely noncommuting variables x=(x1,…,xg)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.

Author information

Contact details are reproduced from the original publication and may be historical.

J. William Helton

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

helton@math.ucsd.edu

Scott McCullough

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

sam@math.ufl.edu

Keywords

  • Matrix convexity
  • free convexity
  • linear matrix inequality
  • noncommutative rational function
  • free rational function

Mathematics Subject Classification

47Axx; 47A63, 47L07, 14P10

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Suggested citation

J. W. Helton, S. McCullough. “Free Convex Sets Defined by Rational Expressions Have LMI Representations.” Journal of Convex Analysis 21 (2014), No. 2, 425–448. https://doi.org/10.68381/jca21022

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Published by Heldermann Verlag, 2014. Rights now held by Banach Press.