On eigenvalues of symmetric matrices with PSD principal submatrices

Abstract

We investigate convexity properties of the set of eigenvalue tuples of $n\times n$ real symmetric matrices, whose all $k\times k$ (where $k\le n$ is fixed) minors are positive semidefinite. It is proven that the set $\lambda ({\mathcal{S}}^{n,k})$ of eigenvalue vectors of all such matrices is star-shaped with respect to the nonnegative orthant ${\mathbb{R}}_{\ge 0}^n$ and not convex already when $(n,k)=(4,2)$.