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arXiv · 0804.0633

Non-Commutative Partial Matrix Convexity

Damon M. Hay·J. William Helton·Adrian Lim·Scott McCullough
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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 $p = L + Λ^T Λ,$ 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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Damon M. Hay, J. William Helton, Adrian Lim, Scott McCullough. 2008-04-03. Non-Commutative Partial Matrix Convexity. https://arxiv.org/abs/0804.0633

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