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

Olga M. Katkova·Anna M. Vishnyakova
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Abstract

The following theorem is proved: Suppose $M = (a_{i,j})$ be a $k \times k$ matrix with positive entries and $a_{i,j}a_{i+1,j+1} > 4\cos ^2 \fracπ{k+1} a_{i,j+1}a_{i+1,j} \quad (1 \leq i \leq k-1, 1 \leq j \leq k-1).$ Then $\det M > 0 .$ The constant $4\cos ^2 \fracπ{k+1}$ 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.

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Olga M. Katkova, Anna M. Vishnyakova. 2005-04-09. On sufficient conditions for the total positivity and for the multiple positivity of matrices. https://arxiv.org/abs/math/0504183

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