Sufficient conditions for total positivity, compounds, and Dodgson condensation

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2025-01-16 DOI:10.1016/j.laa.2025.01.016

Shaun Fallat , Himanshu Gupta , Charles R. Johnson

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引用次数: 0

Abstract

A n-by-n matrix is called totally positive (TP) if all its minors are positive and

T P_{k}

if all of its k-by-k submatrices are TP. For an arbitrary totally positive matrix or

T P_{k}

matrix, we investigate if the rth compound (

1 < r < n

) is in turn TP or

T P_{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

T P_{k}

. We also show that all condensed matrices associated with a TP Hankel matrix are TP.

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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.