Simultaneous kernels of matrix Hadamard powers

TL;DR

This paper extends the understanding of the structure of simultaneous kernels of Hadamard powers from positive semidefinite matrices to a broader class called 3-PMP matrices, revealing new algebraic properties.

Contribution

It generalizes previous results on kernels of Hadamard powers to 3-PMP matrices, broadening the class of matrices for which these structures are understood.

Findings

Extension of kernel structure results to 3-PMP matrices

Development of a new stratification approach for 3-PMP matrices

Broader classification of matrices with zero-one modulus entries

Abstract

In previous work [Adv. Math. 298, pp. 325-368, 2016], the structure of the simultaneous kernels of Hadamard powers of any positive semidefinite matrix were described. Key ingredients in the proof included a novel stratification of the cone of positive semidefinite matrices and a well-known theorem of Hershkowitz, Neumann, and Schneider, which classifies the Hermitian positive semidefinite matrices whose entries are $0$ or $1$ in modulus. In this paper, we show that each of these results extends to a larger class of matrices which we term $3$-PMP (principal minor positive).