振荡矩阵在矩阵与逆矩阵稀疏性下的指数轮廓
Exponent Profiles of Oscillatory Matrices under Matrix and Inverse Sparsity
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中文总结 AI 辅助
本文分类对称振荡矩阵及其逆的半带宽组合，给出指数上界与和的最大值，并刻画基本振荡类中可实现指数向量的充要条件。
中文摘要 AI 辅助
我们分类了对称振荡矩阵A及其逆矩阵的同时半带宽(p,q)。在阶数n≥2时，可能的对为(1,n-1)、(n-1,1)以及所有满足2≤p,q≤n-1的对。每一对都有行列式为1的整数实现。设e_k(A)为使A^m的所有k阶子式为正的最小正整数m。我们证明e_k(A)≤min{max{k,n-p},max{n-k,n-q}}（1≤k<n）。当p+q≥n时，一个实现同时达到所有这些界。对于每个带宽对，确定了∑_{k=1}^{n-1}e_k(A)的最大值。设A=U^TΔU，其中U为单位上三角矩阵，Δ为正对角矩阵。设ℓ(U)为U的正初等上双对角因子个数的最小值。在固定(p,q)下，其最小值为max{n-1,p+q-1}。界∑_{k=1}^{n-1}e_k(A)+ℓ(U)≤n(n-1)/2+n-1在每个因子计数处都是尖锐的；等式使用U、U^2和A的零角子式来刻画。特别地，∑_{k=1}^{n-1}e_k(A)≤n(n-1)/2，等式恰好对基本矩阵成立。在基本振荡类中，在没有对称性假设的情况下，正整数向量(f_1,…,f_{n-1})可实现当且仅当对于1≤k<n，f_k≤max{k,n-k}且f_k+f_{n-k}≥n，并且对于1≤k<n-1，|f_{k+1}-f_k|≤1。显式因子顺序实现每个这样的向量。
英文摘要
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 \(2\le p,q\le n-1\). Every pair has an integer realization of determinant one. Let \(e_k(A)\) be the least positive integer \(m\) for which all minors of order \(k\) of \(A^m\) are positive. We prove \[ e_k(A)\le \min\{\max\{k,n-p\},\max\{n-k,n-q\}\}\qquad(1\le k<n). \] When \(p+q\ge n\), one realization attains all these bounds simultaneously. The maximum of \(\sum_{k=1}^{n-1}e_k(A)\) is determined for every bandwidth pair. Write \(A=U^TΔU\), with \(U\) unit upper triangular and \(Δ\) positive diagonal. Let \(\ell(U)\) be the least number of positive elementary upper bidiagonal factors of \(U\). At fixed \((p,q)\), its minimum is \(\max\{n-1,p+q-1\}\). The bound \(\sum_{k=1}^{n-1}e_k(A)+\ell(U)\le n(n-1)/2+n-1\) is sharp at each factor count; equality is characterized using \(U\), \(U^2\) and the zero corner minors of \(A\). In particular, \(\sum_{k=1}^{n-1}e_k(A)\le n(n-1)/2\), with equality exactly for 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 \(1\le k<n\), and \(\lvert f_{k+1}-f_k\rvert\le1\) for \(1\le k<n-1\). Explicit factor orders realize every such vector.
