Non-Commutative Partial Matrix Convexity
  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. 
Damon M. Hay, J. William Helton, Adrian Lim, and Scott McCullough