A simple and effective factorization procedure to determine total positivity of a matrix

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International Journal of Development Research

A simple and effective factorization procedure to determine total positivity of a matrix

Abstract: 

A factorization procedure for a given totally positive (TP) matrix- a matrix with all positive minors - is introduced. The strategy is to reduce a column to corresponding column of the identity matrix. Factors so obtained are triangular matrices with constant row or column entries. Such matrices and their inverses with simple structures can be constructed using the entries of a given non-zero vector without any computations among the entries. The significance of the factorization is that in general it presents the column and row entries of a matrix as constituted by partial sums of the column and row entries of the factors. Hence for a TP matrix if the entries of its first column and row are in ascending order, this order property will be extended to all its other rows and columns. This order property of entries of the columns and rows will be manifested at each step of the factorization. In an independent way, how factors with all positive entries induce this order property and contribute to the total positivity of a matrix are discussed. Factorization of a given matrix  in the proposed way leads to simple procedures to determine its total positivity. The convenience of the procedure based on set of 2X2 minors which include the first column and set of 2X2 minors which include adjacent rows and columns is that it does not call for factorization of A. It involves n3/3 operations only as against existing n3/2 operations in literature.

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