Improvement On The Methods Of Totally Positive Matrices

ABSTACT:A matrix is said to be totally positive if all its minors are nonnegative. To-tally positive matrices appear not only in a wide range of mathematical subjects includ-ing combinatorics, probability, stochastic process, representation theory, and inverse problems, but also a comprehensive use in economics. Therefore, it is significant to study this type of matrices.It is very difficult to give a good method for determining the totally positive matri-ces. It is impossible to cheek positivity of every minors of the matrix. M.Gasca obtained a elimination method, by study the interpolation formula and elimination, which is a useful method to deal with totally positive matrices. After several steps elimination, we can get a upper triangular matrices, which simplified the process. Through the method above, many decision methods have been appeared to totally positive matrices includ-ing LDU method and QR method which mentioned in this article. A good method to determining a matrix' positivity is to judge its submatrices by the disclose method.This paper depends on matrix LDU factorization, QR factorization and singular value factorization. Combined the methods used in these factorizations we obtain dif-ferent methods to estimate the totally positivity of a matrix.Based on the LDU factor-ization, a new characterization of totally positive matrix is obtained by LU factoriza-tion. The QR factorization on totally positive matrix is introduced. Finally, we give necessary and sufficient conditions that matrices are totally positive by singular value factorization.