Minor-Order Exponent Profiles of Oscillatory Matrices
Abstract
For an n× n oscillatory matrix A, let ek(A) be the least positive integer m for which every minor of order k of Am is positive. We determine the profile (e1(A),…,en(A)). For every nonsingular totally nonnegative matrix, the column index sets of positive entries in each compound row form an interval in the componentwise order. The endpoint maps are order-preserving and compose under multiplication. This yields a two-corner criterion for positivity of all minors of a fixed order. Consequently, ek(A) is the larger of the first positivity times of two remote corner minors. We prove the sharp inequalities ek(A)≤maxk,n−k for 1≤ k<n and |ek+1(A)−ek(A)|≤1 for 1≤ k≤ n−2. For D=diag(1,−1,1,−1,…), the matrix DA−1D is oscillatory and satisfies ek(DA−1D)=en−k(A) for 1≤ k<n; the determinant exponent remains one. The ordered positions of the positive factors in an adjacent bidiagonal factorization determine the profile independently of their values. Each one-sided factor sequence can be represented by a single permutation, giving a finite characterization of all profiles and an integer unimodular realization of each. We also obtain a compatibility condition across minor orders: if 3≤ k≤ n−3 and ek(A)≤2, then ej(A)≤2+⌈ |j−k|/2⌉ for 2≤ j≤ n−2. In particular, ek(A)≤2 implies ek+2(A)≤3 for 3≤ k≤ n−4, and both the constant and this range are sharp.
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BibTeX
@misc{xie2026minororder,
title = {{Minor-Order Exponent Profiles of Oscillatory Matrices}},
author = {Wei Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05787},
eprint = {2609.05787},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05787},
}
RIS
TY - UNPB
TI - Minor-Order Exponent Profiles of Oscillatory Matrices
AU - Wei Xie
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2609.05787
UR - https://arxiv.org/abs/2609.05787
AB - For an n× n oscillatory matrix A, let e_k(A) be the least positive integer m for which every minor of order k of A^m is positive. We determine the profile (e_1(A),…,e_n(A)). For every nonsingular totally nonnegative matrix, the column index sets of positive entries in each compound row form an interval in the componentwise order. The endpoint maps are order-preserving and compose under multiplication. This yields a two-corner criterion for positivity of all minors of a fixed order. Consequently, e_k(A) is the larger of the first positivity times of two remote corner minors. We prove the sharp inequalities e_k(A)≤maxk,n−k for 1≤ k<n and |e_k+1(A)−e_k(A)|≤1 for 1≤ k≤ n−2. For D=diag(1,−1,1,−1,…), the matrix DA^−1D is oscillatory and satisfies e_k(DA^−1D)=e_n−k(A) for 1≤ k<n; the determinant exponent remains one. The ordered positions of the positive factors in an adjacent bidiagonal factorization determine the profile independently of their values. Each one-sided factor sequence can be represented by a single permutation, giving a finite characterization of all profiles and an integer unimodular realization of each. We also obtain a compatibility condition across minor orders: if 3≤ k≤ n−3 and e_k(A)≤2, then e_j(A)≤2+⌈ |j−k|/2⌉ for 2≤ j≤ n−2. In particular, e_k(A)≤2 implies e_k+2(A)≤3 for 3≤ k≤ n−4, and both the constant and this range are sharp.
ER -