Year: 2026
Type: preprint
Abstract: We classify the simultaneous half-bandwidths $p,q$ of a symmetric oscillatory matrix $A$ and its inverse. At order $n\ge2$, the possible pairs are $(1,n-1)$, $(n-1,1)$, and all pairs with $2\le p,q\le n-1$. Let $e_k(A)$ be the least positive integer $m$ for which all minors of order $k$ of $A^m$ are positive, and let $e(A)=\max_{1\le kr\}$ for every integer $r\ge\lfloor(n-1)/2\rfloor$, with the starting point optimal. With $δ=\ell(U)-n+1$, the exponent sum satisfies $δ\le n(n-1)/2-\sum_{k=1}^{n-1}e_k(A) \le\lfloor(n-1)/2\rfloorδ$, 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 $(f_1,\ldots,f_{n-1})$ is realizable if and only if $f_k\le\max\{k,n-k\}$ and $f_k+f_{n-k}\ge n$ for $1\le k
Source: arXiv (Cornell University)
Author: Xie, Wei
Language: English
Topic: Matrix Theory and Algorithms
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214937646