Efficient recognition of totally nonnegative matrix cells

TL;DR

This paper extends the Gasca and Peña criterion for total positivity to all totally nonnegative cells, providing efficient tests based on a small set of minors for matrix membership.

Contribution

It introduces a new, simplified testing method for all totally nonnegative cells using only mp minors, generalizing existing criteria.

Findings

Developed a test for each nonnegative cell using only mp minors

Extended the Gasca and Peña criterion to all nonnegative cells

Provided an efficient, small-set minor test for matrix membership

Abstract

The space of mxp totally nonnegative real matrices has a stratification into totally nonnegative cells. The largest such cell is the space of totally positive matrices. There is a well-known criterion due to Gasca and Pe\~na for testing a real matrix for total positivity. This criterion involves testing mp minors. In contrast, there is no known small set of minors for testing for total nonnegativity. In this paper, we show that for each of the totally nonnegative cells there is a test for membership which only involves mp minors, thus extending the Gasca and Pe\~na result to all totally nonnegative cells.