Some properties of totally positive matrices

AbstractLet A be a real n × n matrix. A is TP (totally positive) if all the minors of A are nonnegative. A has an LU-factorization if A = LU, where L is a lower triangular matrix and U is an upper triangular matrix. The following results are proved: Theorem 1: A is TP iff A has an LU-factorization such that L and U are TP. Theorem 2: If A is TP, then there exist a TP matrix S and a tridiagonal TP matrix T such that (i)TS=SA, and (ii) the matrices A and T have the same eigenvalues. If A is nonsingular, then S is also nonsingular. Theorem 3: If A is an n × n matrix of rank m, then A is TP iff every minor of A formed from any columns β1,…,βp satisfying ∑i=2p |βi−βi−1−1| ⩽ n−m is nonnegative. Theorem 4: If A is a nonsingular lower triangular matrix, then A is TP iff every minor of A formed from consecutive initial columns is nonnegative