Year: 2026
Type: preprint
Abstract: Stone conjectured in 1981 that every fully semimonotone $\mathbf{Q}_0$-matrix is a $\mathbf{P}_0$-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 $A\in\Ezerof$ without using $\Qzero$. Let $\G(A)$ be the graph whose vertices are the index sets $α$ with $\det A_{αα}\neq 0$ (including $α=\emptyset$), 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 $\emptyset$. Consequences: an $\Ezerof$-matrix with no vanishing principal minor is a $\mathbf{P}$-matrix, and any $\Ezerof$-matrix satisfying a simple ``descending'' condition on its nonvanishing minors is a $\mathbf{P}_0$-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-$\mathbf{P}_0$ matrix whose inverse lies in $\Ezerof\cap\Nzero$ and has zero diagonal. Explicit examples show that the component structure can contain negative components, that $\det A>0$ does not force $A\in\Pzero$ in $\Ezerof$, that $\Ezerof$ is not invariant under $A\mapsto A+\varepsilon I$, and that $\Ezerof\cap\Pzero$ contains matrices with $\det A>0$ outside $\Qzero$. Stone's conjecture itself remains open.
Source: arXiv (Cornell University)
Author: Ghosh, Sajal
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204170188