Calculating max-eigenvalues and max-eigenvectors with jumps of matrices

TL;DR

This paper introduces a faster method for computing the maximum eigenvalue and eigenvector of an irreducible non-negative matrix in max-algebra by using matrix mutations, improving upon the traditional power method.

Contribution

It presents a novel, more efficient procedure for calculating max-eigenvalues and eigenvectors in max-algebra through matrix mutation techniques.

Findings

The new method reduces computational complexity compared to the power method.

It accurately computes the maximum cycle geometric mean (eigenvalue).

The approach is applicable to irreducible non-negative matrices in max-algebra.

Abstract

The eigenvalue problem for an irreducible non negative matrix $A=[a_{ij}]$ in the max-algebra is the form $A \otimes x = \lambda x$ where $(A \otimes x)_i = \max (a_{ij}x_j), x=(x_1,x_2, \dots, x_n)^t $ and $\lambda $ refers to maximum cycle geometric mean $\mu (A) $. In this paper we exhibit a method to compute $\mu (A)$ and max-eigenvector by using mutation of matrices. Since the order of power method algorithm is $O(n^3)$, the advantage of this paper present a faster procedure.