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Matrix Inequalities: A Symbolic Procedure to Determine Convexity Automatically
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| Organization: | UC San Diego |
| Department: | Dept. of Mechanical and Aerospace Engineering |
| Organization: | University of California, San Diego |
| Department: | Department of Mathematics |
| Organization: | UC San Diego |
| Department: | Dept. of Mechanical and Aerospace Engineering |
| Organization: | UC San Diego |
| Department: | Dept. of Mathematics |
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| Integral Equations and Operator Theory |
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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.
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noncommutative algebra, Matrix inequalities, Convexity, Computer algebra
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