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 o f a circuit in the weighted directed graph corresponding to A. This th eorem, which we refer to as the max version of the Perron-Frobenius Th eorem, is well-known in the context of matrices over the max algebra a nd also in the context of matrix scalings. In the present work, which is partly expository, we bring out the intimate connection between thi s 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 t o unify the Perron-Frobenius Theorem and its max version are indicated . Some inequalities for mu(A) are proved. (C) 1998 Elsevier Science In c. All rights reserved.