On eigenvalues of symmetric matrices with PSD principal submatrices

TL;DR

This paper studies the geometric properties of eigenvalue sets of symmetric matrices with all principal minors positive semidefinite, revealing star-shapedness and non-convexity in certain cases.

Contribution

It establishes the star-shapedness of eigenvalue sets for matrices with PSD minors and demonstrates non-convexity in specific dimensions.

Findings

Eigenvalue set is star-shaped with respect to the nonnegative orthant.

The set is not convex for (n,k)=(4,2).

Convexity properties depend on matrix size and minor order.

Abstract

We investigate convexity properties of the set of eigenvalue tuples of $n\times n$ real symmetric matrices, whose all $k\times k$ (where $k\leq n$ 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 $\mathbb{R}^n_{\geq 0}$ and not convex already when $(n,k)=(4,2)$.