We consider the (projective) variety $S_{k,n}$ of symmetric $(n \times n)$-matrices of rank at most $k$. This is singular, but by work of Špenko-Van den Bergh it admits a (possibly Brauer twisted) non-commutative crepant resolution $S'_{k,n}$. Following a physical duality proposed by Hori and a conjecture by Kuznetsov, we work out the homological projective duals of these resolved spaces. Modulo taking double covers and Brauer twists according to a prescription based on the parities of $k$ and $n$, we find that $S'_{k,n}$ is HP dual to $S'_{n-k+1,n}$. This is joint work with Ed Segal.