Arveson extreme points span free spectrahedra

• Published in: Mathematische annalen Vol. 375; no. 1-2; pp. 629 - 653
• Main Authors: Evert, Eric, Helton, J. William
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 08.10.2019; Springer Nature B.V
• Subjects: Algorithms; Complex numbers; Computational geometry; Contrarian investing; Convexity; Hulls; Linear matrix inequalities; Manufacturing cells; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix methods; Primary 46L07; 90C22; Secondary 46L07
• ISSN: 0025-5831; 1432-1807
• DOI: 10.1007/s00208-019-01858-9

Let S M n ( R ) g denote g -tuples of n × n real symmetric matrices. Given tuples X = ( X 1 , … , X g ) ∈ S M n 1 ( R ) g and Y = ( Y 1 , … , Y g ) ∈ S M n 2 ( R ) g , a matrix convex combination of X and Y is a sum of the form V 1 ∗ X V 1 + V 2 ∗ Y V 2 V 1 ∗ V 1 + V 2 ∗ V 2 = I n where V 1 : R n → R n 1 and V 2 : R n → R n 2 are contractions. Matrix convex sets are sets which are closed under matrix convex combinations. A key feature of matrix convex combinations is that the g -tuples X ,  Y , and V 1 ∗ X V 1 + V 2 ∗ Y V 2 do not need to have the same size. As a result, matrix convex sets are a dimension free analog of convex sets. While in the classical setting there is only one notion of an extreme point, there are three main notions of extreme points for matrix convex sets: ordinary, matrix, and absolute extreme points. Absolute extreme points are closely related to the classical Arveson boundary. A central goal in the theory of matrix convex sets is to determine if one of these types of extreme points for a matrix convex set minimally recovers the set through matrix convex combinations. This article shows that every real compact matrix convex set which is defined by a linear matrix inequality is the matrix convex hull of its absolute extreme points, and that the absolute extreme points are the minimal set with this property. Furthermore, we give an algorithm which expresses a tuple as a matrix convex combination of absolute extreme points with optimal bounds. Similar results hold when working over the field of complex numbers rather than the reals.