Sign Propagation of Principal Minors in Fully Semimonotone Matrices and Stone's Conjecture
Abstract
Stone conjectured in 1981 that every fully semimonotone Q0-matrix is a P0-matrix; Murthy and Parthasarathy (1995) proved this for orders up to $4$ and for several special classes. We study the principal minors of fully semimonotone matrices A∈Ezerof without using Qzero. Let G(A) be the graph whose vertices are the index sets α with det Aαα≠ 0 (including α=∅), two vertices being adjacent when they differ in exactly one index. We prove that the sign of the principal minor is constant on every connected component of G(A), and positive on the component of ∅. Consequences: an Ezerof-matrix with no vanishing principal minor is a P-matrix, and any Ezerof-matrix satisfying a simple ``descending'' condition on its nonvanishing minors is a P0-matrix. We show that a sign change across two steps through vanishing minors forces strictly positive coupling in the corresponding principal pivotal transform, and that a principal submatrix of minimal order with negative determinant is an almost-P0 matrix whose inverse lies in Ezerof∩Nzero and has zero diagonal. Explicit examples show that the component structure can contain negative components, that det A>0 does not force A∈Pzero in Ezerof, that Ezerof is not invariant under A↦ A+ε I, and that Ezerof∩Pzero contains matrices with det A>0 outside Qzero. Stone's conjecture itself remains open.
Related
- Valuated delta matroids and principal minors of Hermitian matrices
- Principal Minors and Absolute Values
- Ideals of 2-minors of nearly determinantal matrices
- Sum-of-Squares Certificates for Copositive Matrices via Recursive Identities: The de Klerk-Pasechnik Conjecture and Hoffman-Pereira Matrices
- On the heat flow conjecture for random matrices
BibTeX
@misc{ghosh2026sign,
title = {{Sign Propagation of Principal Minors in Fully Semimonotone Matrices and Stone's Conjecture}},
author = {Sajal Ghosh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.22829},
eprint = {2608.22829},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22829},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Sign Propagation of Principal Minors in Fully Semimonotone Matrices and Stone's Conjecture
AU - Sajal Ghosh
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2608.22829
UR - https://arxiv.org/abs/2608.22829
AB - Stone conjectured in 1981 that every fully semimonotone Q_0-matrix is a P_0-matrix; Murthy and Parthasarathy (1995) proved this for orders up to $4$ and for several special classes. We study the principal minors of fully semimonotone matrices A∈Ezerof without using Qzero. Let G(A) be the graph whose vertices are the index sets α with det A_αα≠ 0 (including α=∅), two vertices being adjacent when they differ in exactly one index. We prove that the sign of the principal minor is constant on every connected component of G(A), and positive on the component of ∅. Consequences: an Ezerof-matrix with no vanishing principal minor is a P-matrix, and any Ezerof-matrix satisfying a simple ``descending'' condition on its nonvanishing minors is a P_0-matrix. We show that a sign change across two steps through vanishing minors forces strictly positive coupling in the corresponding principal pivotal transform, and that a principal submatrix of minimal order with negative determinant is an almost-P_0 matrix whose inverse lies in Ezerof∩Nzero and has zero diagonal. Explicit examples show that the component structure can contain negative components, that det A>0 does not force A∈Pzero in Ezerof, that Ezerof is not invariant under A↦ A+ε I, and that Ezerof∩Pzero contains matrices with det A>0 outside Qzero. Stone's conjecture itself remains open.
ER -