Eigenvalues of locally positive semidefinite matrices: Non-convexity and Geometry
Abstract
A real symmetric matrix is called d-locally positive semidefinite if all of its d × d principal submatrices are positive semidefinite. We investigate the spectral geometry of d-locally positive semidefinite matrices. The set of vectors of eigenvalues of d-locally positive semidefinite matrices of size n × n is fully understood and known to be convex when d ∈ {1,n−1,n} \cite{blekherman2022hyperbolic}. In the smallest remaining case n=4, d=2, non-convexity of the set of vectors of eigenvalues was proved in \cite{kozhasov2023eigenvalues}, but even in this case the full description was unknown. We provide a basic semialgebraic description of the set of vectors of eigenvalues for n=4, d=2 by establishing a Fischer-type inequality for $2$-locally positive semidefinite matrices of size 4 × 4 and prove non-convexity for n ≥ 4 and d ∈ {2, n−2}. Non-convexity is established via solving certain non-smooth and non-convex min-max point configuration problems in the complex plane, which could be interesting in themselves. Similar problems were considered in \cite{nesterenko2024submatrices,sengupta2026submatrices} in the context of matrix decomposition and approximation.
Related
- On Completely Positive and Semidefinite Programs for Moment Bounds on Maximum of Random Quadratics
- Comparison theorems for the minimum eigenvalue of a random positive-semidefinite matrix
- Optimal Condition Numbers in Low-Rank Positive Semidefinite Matrix Sensing
- Sharp Edge-Edit Bounds at Every Level for Leaky Positive Semidefinite Forcing
- An optimal pinching theorem on compact minimal submanifolds in the Euclidean spheres via eigenvalues of fundamental matrices
BibTeX
@misc{acevedo2026eigenvalues,
title = {{Eigenvalues of locally positive semidefinite matrices: Non-convexity and Geometry}},
author = {Jose Acevedo and Grigoriy Blekherman and Sebastian Debus and Seokbin Lee and Cordian Riener},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.15444},
eprint = {2608.15444},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.15444},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Eigenvalues of locally positive semidefinite matrices: Non-convexity and Geometry
AU - Jose Acevedo
AU - Grigoriy Blekherman
AU - Sebastian Debus
AU - Seokbin Lee
AU - Cordian Riener
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.15444
UR - https://arxiv.org/abs/2608.15444
AB - A real symmetric matrix is called d-locally positive semidefinite if all of its d × d principal submatrices are positive semidefinite. We investigate the spectral geometry of d-locally positive semidefinite matrices. The set of vectors of eigenvalues of d-locally positive semidefinite matrices of size n × n is fully understood and known to be convex when d ∈ {1,n−1,n} \cite{blekherman2022hyperbolic}. In the smallest remaining case n=4, d=2, non-convexity of the set of vectors of eigenvalues was proved in \cite{kozhasov2023eigenvalues}, but even in this case the full description was unknown. We provide a basic semialgebraic description of the set of vectors of eigenvalues for n=4, d=2 by establishing a Fischer-type inequality for $2$-locally positive semidefinite matrices of size 4 × 4 and prove non-convexity for n ≥ 4 and d ∈ {2, n−2}. Non-convexity is established via solving certain non-smooth and non-convex min-max point configuration problems in the complex plane, which could be interesting in themselves. Similar problems were considered in \cite{nesterenko2024submatrices,sengupta2026submatrices} in the context of matrix decomposition and approximation.
ER -