Exponent Profiles of Oscillatory Matrices under Matrix and Inverse Sparsity
Abstract
We classify the simultaneous half-bandwidths p,q of a symmetric oscillatory matrix A and its inverse. At order n≥2, the possible pairs are (1,n−1), (n−1,1), and all pairs with 2≤ p,q≤ n−1. Let ek(A) be the least positive integer m for which all minors of order k of Am are positive, and let e(A)=max1≤ k<nek(A). Put a=min{p,q} and b=max{p,q}. At every feasible pair of actual half-bandwidths, maxA e(A)= max{⌊n/2⌋, n−a−⌊(max{n−p−q,0})/b⌋}. We also determine the maximum of ∑k=1n−1ek(A) for every pair. When p+q≥ n, one realization maximizes all ek simultaneously. Write A=UTΔU, with U unit upper triangular and totally nonnegative, and Δ positive diagonal. Let ℓ(U) be the least number of elementary upper bidiagonal factors with positive parameters whose product is U. At fixed (p,q), its minimum is max{n−1,p+q−1}. One symmetric integer matrix of determinant one simultaneously attains this minimum, the maximum exponent, and the maximum exponent sum. We prove ℓ(Ur+1)−ℓ(Ur)=#{k:1≤ k<n, ek(A)>r} for every integer r≥⌊(n−1)/2⌋, with the starting point optimal. With δ=ℓ(U)−n+1, the exponent sum satisfies δ≤ n(n−1)/2−∑k=1n−1ek(A) ≤⌊(n−1)/2⌋δ, with both constants sharp. The lower deficit bound is attained at every admissible factor count. Within the symmetric class, a total sum of n(n−1)/2 characterizes basic matrices. In the basic oscillatory class, without a symmetry assumption, a positive integer vector (f1,…,fn−1) is realizable if and only if fk≤max{k,n−k} and fk+fn−k≥ n for 1≤ k<n, and lvert fk+1−fkrvert≤1 for 1≤ k<n−1. Explicit factor orders realize every such vector.
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BibTeX
@misc{xie2026exponent,
title = {{Exponent Profiles of Oscillatory Matrices under Matrix and Inverse Sparsity}},
author = {Wei Xie},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35846},
eprint = {2609.35846},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35846},
}
RIS
TY - UNPB
TI - Exponent Profiles of Oscillatory Matrices under Matrix and Inverse Sparsity
AU - Wei Xie
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.35846
UR - https://arxiv.org/abs/2609.35846
AB - We classify the simultaneous half-bandwidths p,q of a symmetric oscillatory matrix A and its inverse. At order n≥2, the possible pairs are (1,n−1), (n−1,1), and all pairs with 2≤ p,q≤ n−1. Let e_k(A) be the least positive integer m for which all minors of order k of A^m are positive, and let e(A)=max_1≤ k<ne_k(A). Put a=min{p,q} and b=max{p,q}. At every feasible pair of actual half-bandwidths, max_A e(A)= max{⌊n/2⌋, n−a−⌊(max{n−p−q,0})/b⌋}. We also determine the maximum of ∑_k=1^n−1e_k(A) for every pair. When p+q≥ n, one realization maximizes all e_k simultaneously. Write A=U^TΔU, with U unit upper triangular and totally nonnegative, and Δ positive diagonal. Let ℓ(U) be the least number of elementary upper bidiagonal factors with positive parameters whose product is U. At fixed (p,q), its minimum is max{n−1,p+q−1}. One symmetric integer matrix of determinant one simultaneously attains this minimum, the maximum exponent, and the maximum exponent sum. We prove ℓ(U^r+1)−ℓ(U^r)=#{k:1≤ k<n, e_k(A)>r} for every integer r≥⌊(n−1)/2⌋, with the starting point optimal. With δ=ℓ(U)−n+1, the exponent sum satisfies δ≤ n(n−1)/2−∑_k=1^n−1e_k(A) ≤⌊(n−1)/2⌋δ, with both constants sharp. The lower deficit bound is attained at every admissible factor count. Within the symmetric class, a total sum of n(n−1)/2 characterizes basic matrices. In the basic oscillatory class, without a symmetry assumption, a positive integer vector (f_1,…,f_n−1) is realizable if and only if f_k≤max{k,n−k} and f_k+f_n−k≥ n for 1≤ k<n, and lvert f_k+1−f_krvert≤1 for 1≤ k<n−1. Explicit factor orders realize every such vector.
ER -