A factorization of totally nonsingular matrices over a ring with identity

We say that a rectangular matrix over a ring with identity is totally nonsi ngular (TNS) if for all k, all its relevant submatrices, either having k co nsecutive-rows and the first k columns, or k consecutive-columns and the fi rst k rows, are invertible. We prove that a matrix is TNS if and only if it admits a certain factorization with bidiagonal-type factors and certain in vertible entries. This approach generalizes the Loewner-Neville factorizati on usually applied to totally positive matrices. (C) 2000 Elsevier Science Inc. All rights reserved.