Classifying Linear Matrix Inequalities via Abstract Operator Systems

TL;DR

This paper explores how properties of abstract operator systems can classify linear matrix inequalities, providing complete descriptions for certain cones and characterizing isometries between matrix algebras.

Contribution

It introduces a systematic approach to classifying LMIs using operator system theory, especially for polyhedral cones, Lorentz cone, and positive semidefinite matrices.

Findings

Complete descriptions of LMI definitions for specific cones

Characterization of isometries between matrix algebras

Application of operator system theory to classify LMIs

Abstract

We systematically study how properties of abstract operator systems help classifying linear matrix inequality definitions of sets. Our main focus is on polyhedral cones, the 3-dimensional Lorentz cone, where we can completely describe all defining linear matrix inequalities, and on the cone of positive semidefinite matrices. Here we use results on isometries between matrix algebras to describe linear matrix inequality definitions of relatively small size. We conversely use the theory of operator systems to characterize special such isometries.