Non-Commutative Partial Matrix Convexity

2008/04/03 by Damon M. Hay, Hay, Damon M., J. William Helton +6
Computer Science · Mathematics · #Advanced Optimization Algorithms Research #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Matrix Theory and Algorithms #Optimization and Control (math.OC) #math.FA #math.OC

paper · pdf · doi:10.48550/arxiv.0804.0633

24 pages

arxiv created 2008/04/03 · openalex publication_date 2008/04/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let p be a polynomial in the non-commuting variables (a,x)=(a₁,...,a_ga,x₁,...,x_gx). If p is convex in the variables x, then p has degree two in x and moreover, p has the form p = L + Λ^T Λ, where L has degree at most one in x and Λ is a (column) vector which is linear in x, so that Λ^TΛ is a both sum of squares and homogeneous of degree two. Of course the converse is true also. Further results involving various convexity hypotheses on the x and a variables separately are presented.

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