0
\end{smallmatrix}\right]$ on the diagonal.

Let $X$ be the block diagonal matrix with $m$ blocks $\left[\begin{smallmatrix}
0&1\\
1 & 0
\end{smallmatrix}\right]$ on the diagonal. Then $X^tBX=B^{-1}$. Hence with $Y=PXP^t$, we have $Y^tSY=S^{-1}$. Thus $${\sigma}(SY)SY=S (SY)^t S^{-1}SY= SY^tS^tY=SY^t(-S)Y=-SS^{-1}=-I_n.
\qedhere$$

Positive noncommutative polynomials {#sec5}
===================================

Algebras of generic matrices with involution {#sec:generic}
--------------------------------------------

After studying the PS Conjecture in the setting of central simple algebras with involution, we proceed to interpret these results as well as Theorem \[thm:NCh17\] for non-dimensionfree positivity of noncommutative (NC) polynomials.

Motivated by problems in optimization and control theory, Helton [@Hel] proved that a symmetric real or complex NC polynomial, all of whose images under algebra $*$-homomorphisms into $M_n({\mathbb{R}})$, $n\in{\mathbb{N}}$, are positive semidefinite (i.e., a dimensionfree positive NC polynomial), is a sum of hermitian squares. What we are interested in, is positivity under evaluations in $M_n({\mathbb{R}})$ for a *fixed* $n$.

To tackle this problem we introduce the language of generic matrices [@Row]. Verifying a condition on evaluations of an NC polynomial in the algebra of $n\times n$ matrices is often conveniently done in the algebra of generic matrices. In this subsection we recall the definition of generic matrices with involution, while our main result on positive NC polynomials 