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 \(n\ge2\), 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.