Eigenvalue Optimization of Polynomials in Non-commuting Variables
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In Sect. 1.6 we introduced a natural notion of positivity that corresponds exactly to nc polynomials that are SOHS. Recall that an nc polynomial is positive semidefinite if it yields a positive semidefinite matrix when we replace the letters (variables) in the polynomial by symmetric matrices of the same order. Helton’s Theorem 1.30 implies that positive semidefinite polynomials are exactly the SOHS polynomials, the set of which we denoted by Σ2.
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