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

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.

We find there is a gap in the proof of main theorem which can not be corrected easily and the correction may destroy other corollaries. So we prefer to withdraw the paper