Generalized Oscillatory Matrices

Abstract

Gantmacher and Krein from their investigations on oscillations of mechanical systems, defined an important class of matrices, called oscillatory matrices. An n×n totally nonnegative matrix A is called oscillatory if some positive integral power of it is totally positive. Besides this, we note that some of properties of the oscillatory matrix studied by Ando can be extended to rectangular matrices. We note that Fallat in [4] analyzes a classical criterion obtained by Gantmacher and Krein for determining when a totally nonnegative matrix is oscillatory and he shows a new proof of this criterion by using bidiagonal factorizations of invertible totally nonnegative matrices and using the associated weighted planar network. In this paper, we extend the oscillatory matrices to rectangular matrices and we obtain a characterization of this class of matrices by incorporating bidiagonal factorizations. On the other hand, the totally nonnegative matrices in double echelon form, introduced by Crans, Fallat and Johnson...