Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions

Abstract

An operator C on a Hilbert space H dilates to an operator T on a Hilbert space K if there is an isometry V : H → K such that C = V ∗TV . A main result of this paper is, for a positive integer d, the simultaneous dilation, up to a sharp factor θ(d), expressed as a ratio of Γ functions for d even, of all d× d symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space. Dilating to commuting operators has consequences for the theory of linear matrix inequalities (LMIs). Given a tuple A = (A1, . . . , Ag) of ν×ν symmetric matrices, L(x) := I− ∑ Ajxj is a monic linear pencil of size ν. The solution set SL of the corresponding linear matrix inequality, consisting of those x ∈ R for which L(x) 0, is a spectrahedron. The set DL of tuples X = (X1, . . . , Xg) of symmetric matrices (of the same size) for which L(X) := I − ∑ Aj ⊗Xj is positive semidefinite, is a free spectrahedron. It is shown that any tuple X of d×d symmetric matrices in a bounded free spectrahedron DL dilates, up to a scale factor depending only on d, to a tuple T of commuting self-adjoint operators with joint spectrum in the corresponding spectrahedron SL. From another viewpoint, the scale factor measures the extent that a positive map can fail to be completely positive. Given another monic linear pencil L, the inclusion DL ⊆ DL obviously implies the inclusion SL ⊆ SL and thus can be thought of as its free relaxation. Determining if one free spectrahedron contains another can be done by solving an explicit LMI and is thus computationally tractable. The scale factor for commutative dilation of DL gives a precise measure of the worst case error inherent in the free relaxation, over all monic linear pencils L of size d. The set C of g-tuples of symmetric matrices of norm at most one is an example of a free spectrahedron known as the free cube and its associated spectrahedron is the cube [−1, 1]. The free relaxation of the the NP-hard inclusion problem [−1, 1] ⊆ SL was introduced by Ben-Tal and Nemirovski. They obtained the lower bound θ(d), expressed as the solution of an optimization problem over diagonal matrices of trace norm 1, for the divergence between the original and relaxed problem. The result here on simultaneous dilations of contractions proves this bound is sharp. Determining an analytic formula for θ(d) produces, as a by-product, new probabilistic results for the binomial and beta distributions. Date: August 24, 2015. 2010 Mathematics Subject Classification. 47A20, 46L07, 13J30 (Primary); 60E05, 33B15, 90C22 (Secondary).