On sufficient conditions for the total positivity and for the multiple positivity of matrices

Abstract

The following theorem is proved: Suppose $M=(a_{i,j})$ be a $k\times k$ matrix with positive entries and $a_{i,j}{a}_{i+1,j+1}>4{\cos }^2\frac{\pi }{k+1}{a}_{i,j+1}{a}_{i+1,j}(1\le i\le k-1,1\le j\le k-1).$ Then $\det M>0.$ The constant $4{\cos }^2\frac{\pi }{k+1}$ in this Theorem is sharp. A few other results concerning totally positive and multiply positive matrices are obtained. Keywords: Multiply positive matrix; Totally positive matrix; Strictly totally positive matrix; Toeplitz matrix; Hankel matrix; Pólya frequency sequence.

15 pages