Totally Positive Matrices And Neville Elimination

A real matrix is said to be totally positive (TP) if all its minors are nonnegative, if all its minors are positive, which is said to be strictly totally positive (STP). Such matrices appear in a wide range of mathematical subjects including combinatorics, probability, stochastic process, representation theory, and inverse problems. M. Gasca and G. Mahlbach obtained a elimination method, called Neville elimination, by studying the interpolation formula and elimination techniques. Although this method is not more popular than Gaussian elimination, which is a useful method to deal with TP and STP matrices. The essence of Neville elimination is to make zeros in a column of matrix by adding to each row a multiple of the previous one. Fox an order of n matrix, after n steps of elimination, we can get a upper triangular matrix in the end. In this paper, we will use Neville elimination to do some researches on TP and STP matrices. The whole paper can be divided into the following four Chapters:In Chapter 1, we give a review of history of TP and STP matrices and some important results of TP and STP matrices obtained ago by using Neville elimination.In Chapter 2, we will use Neville elimination put by Gassca to do some researches on rectangular STP matrices, and prove that a rectangular matrix is STP if and only if it can be factorized as a product of two diagonal TP matrices. We also call this factorization as Neville factorization. We also survey the factorizations of rectangular TP matrices and any real matrices obtained by Gasso.Chapter 3 mainly discusses a class of generalized matrices which have the consecutive-column property and the consecutive-row property compared with TP and STP matrices. Here we say that a matrix over a ring with identity has the consecutive-column (CC) property if for all k, all its relevant submatrices having k consecutive rows and the first k columns are invertible. Similarly, we say that a matrix has the consecutive-row (CR) property if for all k, all its relevant submatrices having k consecutive columns and the first k rows are invertible. Fiedler showed that, analogously to TP and STP matrices, a matrix has CC and CR properties if and only if it admits a factorization as a product of bidiagonal matrices with invertible entries. We give a simple proof for the rectangular matrix with CC...