Example of a 6-by-6 Matrix with Different Tropical and Kapranov Ranks

TL;DR

This paper presents a specific 6-by-6 matrix demonstrating a difference between tropical and Kapranov ranks, addressing a previously open question in matrix rank theory.

Contribution

It provides the first explicit example of a matrix with tropical rank 4 and Kapranov rank 5, clarifying the relationship between these ranks.

Findings

First known example of such a matrix

Tropical rank is less than Kapranov rank in this case

Answers an open question in matrix rank theory

Abstract

We provide an example of a 6-by-6 matrix with tropical rank equal to 4 and Kapranov rank equal to 5. This answers a question asked by M. Chan, A. Jensen, and E. Rubei.