On generators of bounded ratios of minors for totally positive matrices

Details On generators of bounded ratios of minors for totally positive matrices On generators of bounded ratios of minors for totally positive matrices

2008-04-21

arxiv.org/abs/0804.3226v1

Linear Algebra and Its Applications (2008) Vol 428/7 pp 1664-1684

Mathematics

Rings and Algebras

27 pages, 2 figures

Scientific paper

10.1016/j.laa.2007.10.011

We provide a method for factoring all bounded ratios of the form $$\det A(I_1|I_1')\det A(I_2|I_2')/\det A(J_1|J_1')\det A(J_2|J_2')$$ where $A$ is a totally positive matrix, into a product of more elementary ratios each of which is bounded by 1, thus giving a new proof of Skandera's result. The approach we use generalizes the one employed by Fallat et al. in their work on principal minors. We also obtain a new necessary condition for a ratio to be bounded for the case of non-principal minors.

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