Triangulation of diagonally dominant min-plus matrices

Abstract

The min-plus algebra is a commutative semiring with two operations: addition \(\varvec{a} \oplus \varvec{b := \min (a,b)}\) and multiplication \(\varvec{a} \otimes \varvec{b := a + b}\). In this paper, we discuss a min-plus algebraic counterpart of matrix diagonalization in conventional linear algebra. Due to the absence of subtraction in the min-plus algebra, few matrices admit such a canonical form. Instead, we consider triangulation of min-plus matrices in terms of algebraic eigenvectors, which is an extended concept of usual eigenvectors. We deal with two types of min-plus matrices: strongly diagonally dominant (SDD) and nearly diagonally dominant (NDD) matrices. For an SDD matrix, the roots of the characteristic polynomial coincide with its diagonal entries. On the other hand, for an NDD matrix, the roots except for the maximum one appear in diagonal entries. We show that SDD matrices admit upper triangulation whose diagonal entries are algebraic eigenvalues, while NDD matrices admit block upper triangulation. We exhibit applications of triangulation of min-plus matrices to traffic flow models.