Random spectrahedra

Spectrahedra are affine-linear sections of the cone $\mathcal{P}_n$ of positive semidefinite symmetric $n\times n$-matrices. We consider random spectrahedra that are obtained by intersecting~$\mathcal{P}_n$ with the affine-linear space $\mathbf{1} + V$, where $\mathbf{1}$ is the identity matrix and $V$ is an $\ell$-dimensional linear space that is chosen from the unique orthogonally invariant probability measure on the Grassmanian of $\ell$-planes in the space of $n\times n$ real symmetric matrices (endowed with the Frobenius inner product). Motivated by applications, for $\ell=3$ we relate the average number $\mathbb{E} \sigma_n$ of singular points on the boundary of a three-dimensional spectrahedron to the volume of the set of symmetric matrices whose two smallest eigenvalues coincide. In the case of quartic spectrahedra ($n=4$) we show that $\mathbb{E} \sigma_4 = 6-\frac{4}{\sqrt{3}}$. Moreover, we prove that the average number $\mathbb{E} \rho_n$ of singular points on the real variety of singular matrices in $\mathbf{1} + V$ is $n(n-1)$. This quantity is related to the volume of the variety of real symmetric matrices with repeated eigenvalues. Furthermore, we compute the asymptotics of the volume and the volume of the boundary of a random spectrahedron.