On the power method in max algebra

Abstract

The eigenvalue problem for an irreducible nonnegative matrix $A = [a_{ij}]$ in the max algebra system is $A\\otimes x = \\lambda x$, where $(A \\otimes x)_i ={\\mathop{{\\max}_j}}(a_{ij}x_j)$ and $\\lambda$ turns out to be the maximum circuit geometric mean, $\\mu(A)$. A power method algorithm is given to compute $\\mu(A)$ and eigenvector $x$. The algorithm is developed by using results on the convergence of max powers of $A$, which are proved using nonnegative matrix theory. In contrast to an algorithm developed in [4], this new method works for any irreducible nonnegative $A$, and calculates eigenvectors in a simpler and more efficient way. Some asymptotic formulas relating $\\mu(A)$, the spectral radius and norms are also given.