A remark on oscillatory matrices
The class of oscillatory matrices introduced by \textit{F. P. Gantmacher} and \textit{M. G. Krein} [Oscillation matrices and kernels and small vibrations of mechanical systems (2002; Zbl 1002.74002)] is considered. An \(n\times n\) matrix \(A\) is called totally nonnegative (resp. totally positive) if any minor of \(A\) is nonnegative (resp. positive). A square totally nonnegative matrix \(A\) is called oscillatory if some positive integral power of \(A\) is totally positive. The classical criterion due to Gantmacher and Krein states that an \(n\times n\) totally nonnegative matrix \(A=[a_{ij}]\) is oscillatory if and only if \(A\) is nonsingular and \(a_{i,i+1}>0,\;a_{i+1,i}>0\) for \(i=1,2,\dots,n-1\). The aim of this paper is to shed new light on this criterion by a new proof which incorporates bidiagonal factorizations of totally nonnegative matrices and some planar diagrams associated with such factorizations. Some key ideas are emphasized, which underline the combinatorial character of this new proof.
scientific article; zbMATH DE number 2126347