Generalized totally positive matrices

Abstract

We say that a rectangular matrix over a (in general, noncommutative) ring with identity having a positive part is generalized totally positive (GTP) if in all nested sequences of so-called relevant submatrices, the Schur complements are positive. Here, a relevant submatrix is such either having k consecutive rows and the first k columns, or k consecutive columns and the first k rows. This notion generalizes the usual totally positive matrices. We prove e.g. that a square matrix is GTP if and only if it admits a certain factorization with bidiagonal-type factors and certain invertible entries. Also, the product of square GTP-matrices of the same order is again a GTP-matrix, and its inverse has the checkerboard-sign property.