Matrix Inequalities: a Symbolic Procedure to Determine Convexity Automatically

This paper gives a theory of noncommutative functions which results in an algorithm for determining where they are “matrix convex”. Of independent interest is a theory of noncommutative quadratic functions and the resulting algorithm which calculates the region where they are “matrix positive semidefinite”. This is accomplished via a theorem on writing noncommutative quadratic functions with noncommutative rational coefficients as a weighted sum of squares. Also the paper gives an LDU algorithm for matrices with noncommutative entries and conditions guaranteeing when it is successful. The motivation for the paper comes from systems engineering. Inequalities involving polynomials in matrices and their inverses and associated optimization problems have become very important in engineering. When these polynomials are “matrix convex” interior point methods apply directly. A difficulty is that often an engineering problem presents a matrix polynomial whose convexity takes considerable skill, time, and luck to determine. Typically this is done by looking at a formula and recognizing “complicated patterns involving Schur complements”; a tricky hit or miss procedure. Certainly computer assistance in determining convexity would be valuable. This paper describes a symbolic algorithm and software which represent a beginning along these lines. Our procedure proceeds automatically and avoids completely Schur complement wizardry. The algorithms described here have been implemented under Mathematica and the noncommutative algebra package NCAlgebra. They will soon be available at www.math.ucsd.edu/∼ncalg. Examples presented in this article illustrate some of this software.