On eigenvalues of symmetric matrices with PSD principal submatrices

We investigate convexity properties of the set of eigenvalue tuples of n x n real symmetric matrices, all of whose k x k (where k < n is fixed) minors are positive semidefinite. We prove that the set lambda(Sn,k) of eigenvalue vectors of all such matrices is star-shaped with respect to the nonnegative orthant Rn >0 and not convex already when (n, k) = (4, 2). We also show that k is the smallest integer such that Sn,kis a linear projection of a set described by linear matrix inequalities of size k. (c) 2023 Elsevier Ltd. All rights reserved.