On the power method in max algebra Original Research Article

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.