Matrix Representations for Positive Noncommutative Polynomials |
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Authors: | J William Helton Scott McCullough Mihai Putinar |
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Institution: | (1) Department of Mathematics, University of California, San Diego, CA 92093, USA;(2) Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, USA;(3) Department of Mathematics, University of California, Santa Barbara, CA 93106, USA |
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Abstract: | In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict
positivity fails.
Specifically, we treat a ``symmetric' polynomial q(x, h) in noncommuting variables, {x1, . . . , } and {h1, . . . , } for which q(X,H) is positive semidefinite whenever
are tuples of selfadjoint matrices with ||Xj|| ≤ 1 but Hj unconstrained. The representation we obtain is a Gram representation in the variables h
where Pq is a symmetric matrix whose entries are noncommutative polynomials only in x and V is a ``vector' whose entries are polynomials in both x and h. We show that one can choose Pq such that the matrix Pq(X) is positive semidefinite for all ||Xj|| ≤ 1. The representation covers sum of square results (Am. Math. (to appear); Linear Algebra Appl. 326 (2001), 193–203; Non commutative Sums of Squares, preprint]) when gx = 0. Also it allows for arbitrary degree in h, rather than degree two, in the main result of Matrix Inequalities: A Symbolic Procedure to Determine Convexity Automatically
to appear IOET July 2003] when restricted to x-domains of the type ||Xj|| ≤ 1.
Partially supported by NSF, DARPA and Ford Motor Co.
Partially supported by NSF grant DMS-0140112
Partially supported by NSF grant DMS-0100367 |
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