On the Rank of a Tropical Matrix

Summary

This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank. We demonstrate how these notions arise naturally in polyhedral and algebraic geometry, and we show that they differ in general. Realizability of matroids plays a crucial role here. Connections to optimization are also discussed.

1. Introduction

The rank of a matrix M is one of the most important notions in linear algebra. This number can be defined in many different ways. In particular, the following three definitions are equivalent:

• • The rank of M is the smallest integer r for which M can be written as the sum of r rank one matrices. A matrix has rank 1 if it is the product of a column vector and a row vector.

• • The rank of M is the smallest dimension of any linear space containing the columns of M.

• • The rank of M is the largest integer r such that M has a nonsingular r x r minor.

Our objective is to examine these familiar definitions over an algebraic structure which has no additive inverses.