Free Convex Sets Defined by Rational Expressions Have LMI Representations

Suppose

is a symmetric matrix whose entries are polynomials in freely noncommutative variables and

p(0) p ( 0 )

is positive definite. Let

{\mathcal D}_p D p

denote the component of zero of the set of those

g g

-tuples

X=(X_1,\dots,X_g) X = ( X 1 , … , 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

{\mathcal D}_p D p

is convex and bounded, then

{\mathcal D}_p D 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

r r

for a symmetric free matrix-valued rational function

\mathfrak{r} r

in

g g

freely noncommuting variables

x=(x_1,\dots,x_g) x = ( x 1 , … , 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.