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

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

2005-04-09

arxiv.org/abs/math/0504183v1

Mathematics

Rings and Algebras

15 pages

Scientific paper

The following theorem is proved: Suppose $M = (a_{i,j})$ be a $k \times k$ matrix with positive entries and $a_{i,j}a_{i+1,j+1} > 4\cos ^2 \frac{\pi}{k+1} a_{i,j+1}a_{i+1,j} \quad (1 \leq i \leq k-1, 1 \leq j \leq k-1).$ 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; P\'olya frequency sequence.

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