A max version of the Perron-Frobenius theorem

If A is an n x n nonnegative, irreducible matrix, then there exists mu(A) > 0, and a positive vector x such that max(j)a(ij)x(j) = mu(A)x(i), i = 1, 2, ..., n. Furthermore, mu(A) is the maximum geometric mean of a circuit in the weighted directed graph corresponding to A. This theorem, which we refer to as the max version of the Perron-Frobenius Theorem, is well-known in the context of matrices over the max algebra and also in the context of matrix scalings. In the present work, which is partly expository, we bring out the intimate connection between this result and the Perron-Frobenius theory. We present several proofs of the result, some of which use the Perron-Frobenius Theorem. Structure of max eigenvalues and max eigenvectors is described. Possible ways to unify the Perron-Frobenius Theorem and its max version are indicated. Some inequalities for mu(A) are proved. (C) 1998 Elsevier Science Inc. All rights reserved.