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arXiv · 2405.06069

Sufficient conditions for total positivity, compounds, and Dodgson condensation

Shaun Fallat·Himanshu Gupta·Charles R. Johnson
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Abstract

A $n$-by-$n$ matrix is called totally positive ($TP$) if all its minors are positive and $TP_k$ if all of its $k$-by-$k$ submatrices are $TP$. For an arbitrary totally positive matrix or $TP_k$ matrix, we investigate if the $r$th compound ($1<r<n$) is in turn $TP$ or $TP_k$, and demonstrate a strong negative resolution in general. Focus is then shifted to Dodgson's algorithm for calculating the determinant of a generic matrix, and we analyze whether the associated condensed matrices are possibly totally positive or $TP_k$. We also show that all condensed matrices associated with a $TP$ Hankel matrix are $TP$.

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Shaun Fallat, Himanshu Gupta, Charles R. Johnson. 2024-05-09. Sufficient conditions for total positivity, compounds, and Dodgson condensation. https://arxiv.org/abs/2405.06069

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