On eigenvalues of symmetric matrices with PSD principal submatrices

2021/03/29 by Khazhgali Kozhasov, Kozhasov, Khazhgali
Computer Science · Engineering · Mathematics · #14P10 #15A18 #15B57 #90C22 #Algebraic Geometry (math.AG) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Optimization and Control (math.OC) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2103.15811

openalex publication_date 2021/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We investigate convexity properties of the set of eigenvalue tuples of n× n real symmetric matrices, whose all k× k (where k≤ n is fixed) minors are positive semidefinite. It is proven that the set λ(S^n,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).

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