Non-Commutative Partial Matrix Convexity

TL;DR

This paper characterizes non-commutative polynomials convex in certain variables, showing they must be quadratic sums of linear and degree-one parts, with implications for convexity in non-commutative algebra.

Contribution

It provides a complete structural characterization of polynomials convex in non-commuting variables, revealing their degree and form, and explores convexity conditions in different variable sets.

Findings

Convex polynomials in non-commuting variables are quadratic sums of linear parts.

Such polynomials have degree at most two in the convex variables.

The form involves a linear part plus a sum of squares.

Abstract

Let $p$ be a polynomial in the non-commuting variables $(a,x)=(a_1,...,a_{g_a},x_1,...,x_{g_x})$. If $p$ is convex in the variables $x$, then $p$ has degree two in $x$ and moreover, $p$ has the form $p = L + \Lambda ^T \Lambda,$ where $L$ has degree at most one in $x$ and $\Lambda$ is a (column) vector which is linear in $x,$ so that $\Lambda^T\Lambda$ 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.