Extensions of S-Lemma for Noncommutative Polynomials

TL;DR

This paper extends the classical S-lemma to noncommutative polynomials, establishing conditions for positive semidefiniteness across various noncommutative polynomial classes.

Contribution

It generalizes the S-lemma to three types of noncommutative polynomials, broadening its applicability in noncommutative algebra.

Findings

Symmetric quadratic homogeneous matrix-valued polynomials are positive semidefinite iff their coefficient matrices are positive semidefinite.

Extended the S-lemma to noncommutative polynomials with real coefficients.

Extended the S-lemma to matrix-valued and hereditary noncommutative polynomials.

Abstract

We consider the problem of extending the classical S-lemma from commutative case to noncommutative cases. We show that a symmetric quadratic homogeneous matrix-valued polynomial is positive semidefinite if and only if its coefficient matrix is positive semidefinite. Then we extend the S-lemma to three kinds of noncommutative polynomials: noncommutative polynomials whose coefficients are real numbers, matrix-valued noncommutative polynomials and hereditary polynomials.