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Matrix Inequalities: A Symbolic Procedure to Determine Convexity Automatically

Juan F. Camino
Organization: UC San Diego
Department: Dept. of Mechanical and Aerospace Engineering
J. William Helton
Organization: University of California, San Diego
Department: Department of Mathematics
URL: http://math.ucsd.edu/~helton/
Robert E. Skelton
Organization: UC San Diego
Department: Dept. of Mechanical and Aerospace Engineering
Jieping Ye
Organization: UC San Diego
Department: Dept. of Mathematics
Journal / Anthology

Integral Equations and Operator Theory
Year: 2003
Volume: 46
Issue: 4
Page range: 399-454

This paper presents 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 algortihm which calculates the region where they are "matrix positive". This is accomplished via a theorem (a type of Positivstellansatz) on writing noncommutative quadratic functions with noncommutative rational coefficients as a weighted sum of squares. furthermore the paper gives an LDU algorithm for matrices with noncommutative entries and conditions guaranteeing that the decomposition is successful.

The motivation for the paper comes from ssytems 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 engineeringn problem presents a matrix polynomial whose convexity takes considerable skill, time, and luck to determine. Typically this si doen 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 in adition to theory, describes a symbolic algorithm and software which represent a beginning along these lines.

The algortithms described here have been implemented under Mathematica and the noncommutative algebra package NCAlgebra. Examples presented iin this article illustrate its use.

*Mathematics > Algebra > Linear Algebra

noncommutative algebra, Matrix inequalities, Convexity, Computer algebra
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*Noncommutative Algebra Package and Systems   [in MathSource: Packages and Programs]