Nonnegative minors of minor matrices

TL;DR

This paper proves that 2x2 minors of minor matrices derived from totally nonnegative matrices remain nonnegative, using combinatorial network interpretations to establish their properties.

Contribution

It introduces a combinatorial framework linking minors of minor matrices to path weights in networks, extending total nonnegativity properties.

Findings

2x2 minors of minor matrices are nonnegative

Combinatorial interpretation via path weights

Extension of total nonnegativity properties

Abstract

Using the relationship between totally nonnegative matrices and directed acyclic weighted planar networks, we show that $2\times 2$ minors of minor matrices of totally nonnegative matrices are also nonnegative. We give a combinatorial interpretation for the minors of minor matrices in terms of the weights of families of paths in a network.