# Solving Linear Systems over Idempotent Semifields through $LU$-factorization

## TL;DR

This paper develops an $LU$-factorization method for square matrices over idempotent semifields, especially max-plus algebra, enabling solutions for linear systems in this algebraic setting.

## Contribution

It introduces a new $LU$-factorization technique for idempotent semifields and extends classical methods to this algebraic context.

## Key findings

- Conditions for $LU$-factorization of matrices over idempotent semifields
- Method for solving linear systems using $LU$-factorization in this setting
- Provided Maple procedures for practical implementation

## Abstract

In this paper, we introduce and analyze a new $LU$-factorization technique for square matrices over idempotent semifields. In particular, more emphasis is put on "max-plus" algebra here, but the work is extended to other idempotent semifields as well. We first determine the conditions under which a square matrix has $LU$ factors. Next, using this technique, we propose a method for solving square linear systems of equations whose system matrices are $LU$-factorizable. We also give conditions for an $LU$-factorizable system to have solutions. This work is an extension of similar techniques over fields. Maple procedures for this $LU$-factorization are also included.