Generalized oscillatory matrices (Q1864962)

The paper is a continuation of the recent series of the authors' works [see for example \textit{M. Fiedler} and \textit{T. L. Markham}, Linear Algebra Appl. 345, 9-28 (2002; Zbl 0995.15016) and references therein] on the theory of totally nonnegative matrices over noncommutative rings. Let \(R\) be a noncommutative ring with the identity 1 and positive part \(R^+\) which is defined as follows: 1. if \(a,b\in R^+\) then \(a+b,ab\in R^+\); 2. if \(a\in R^+\) then \(a\) is invertible and \(a^{-1}\in R^+\). An LB-matrix (resp., UB-matrix) is a square bidiagonal matrix whose diagonal entries are either \(1\) or \(0\), and the subdiagonal (resp., superdiagonal) entries are either zero or positive elements of \(R\). A D-matrix is a diagonal matrix whose diagonal entries are either zero or positive. A B-matrix is either a \(k\times (k+1)\) matrix the diagonal entries of which are either \(1\) or \(0\) and the entries in the parallel second diagonal are either positive or zero, or a transpose of such matrix. A generalized totally nonnegative (GTN) matrix over \(R\) is a product of LB-, UB-, B-, and D-matrices in any order. A generalized invertible totally nonnegative (GITN) matrix over \(R\) is any finite product of invertible LB-, UB-, and D-matrices in any order. A square matrix over \(R\) is called generalized oscillatory (GO) if it is GTN and some power of it is generalized totally positive. Basic generalized oscillatory (BGO) matrices are defined as GITN-matrices of a special canonical form. It is proved that BGO-matrices are GO. The authors show that each BGO-matrix has both subdiagonal and superdiagonal rank one. A characterization theorem for an \(n\times n\)-matrix to be BGO is provided which implies that an \(n\times n\) BGO-matrix is uniquely determined by \(3n-2\) positive parameters and two increasing sequences of positive integers ending by \(n-1\). Some characterizations of GO matrices are obtained. In particular, it is proved that a matrix \(A\in M_n(R)\) is GO if and only if \(A\) is GITN and irreducible. The connections with the ordering on invertible GTN-matrices, defined in previous works of the authors, are investigated. Several examples and open problems are presented.