Characterizing total negativity and testing their interval hulls

by   Projesh Nath Choudhury, et al.

A matrix is called totally negative (totally non-positive) of order k, if all its minors of size at most k are negative (non-positive). The objective of this article is to provide several novel characterizations of total negativity via the (a) sign non-reversal property, (b) variation diminishing property, and (c) Linear Complementarity Problem. More strongly, each of these three characterizations uses a single test vector. As an application of the sign non-reversal property, we study the interval hull of two rectangular matrices. In particular, we identify two matrices C^±(A,B) in the interval hull of matrices A and B that test total negativity of order k, simultaneously for the entire interval hull. We also show analogous characterizations for totally non-positive matrices. These novel characterizations may be considered similar in spirit to fundamental results characterizing totally positive matrices by Brown–Johnstone–MacGibbon [J. Amer. Statist. Assoc. 1981] (see also Gantmacher–Krein, 1950), Choudhury–Kannan–Khare [Bull. London Math. Soc., in press] and Choudhury [2021 preprint]. Finally using a 1950 result of Gantmacher–Krein, we show that totally negative/non-positive matrices can not be detected by (single) test vectors from orthants other than the open bi-orthant that have coordinates with alternating signs, via the sign non-reversal property or the variation diminishing property.



There are no comments yet.


page 1

page 2

page 3

page 4


Sign non-reversal property for totally positive matrices and testing total positivity on their interval hull

We establish a novel characterization of totally positive matrices via a...

Interval hulls of N-matrices and almost P-matrices

We establish a characterization of almost P-matrices via a sign non-reve...

Variation diminishing linear time-invariant systems

This paper studies the variation diminishing property of k-positive syst...

Misuse of the sign test in narrative synthesis of evidence

In narrative synthesis of evidence, it can be the case that the only qua...

Estimation of positive definite M-matrices and structure learning for attractive Gaussian Markov Random fields

Consider a random vector with finite second moments. If its precision ma...

Simple, fast and accurate evaluation of the action of the exponential of a rate matrix on a probability vector

Given a time-homogeneous, finite-statespace Markov chain with a rate mat...

Positive results from UK single gene testing for SARS-COV-2 may be inconclusive, negative or detecting past infections

The UK Office for National Statistics (ONS) publish a regular infection ...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.