Abstract :
In the max algebra system, for an n × n nonnegative matrix A = [aij] the eigenequation for max eigenvalue λ and corresponding max eigenvector x is A circle times operator x = λx, where [A circle times operator x]i = max1less-than-or-equals, slantjless-than-or-equals, slantnaijxj and μ(A) is the maximum circuit geometric mean. It is shown that the following conditions are mutually equivalent: (i) ηshort parallel·short parallel(A) < 1, for some norm short parallel·short parallel on image; (ii) image; (iii) μ(A) < 1; (iv) image, where ηshort parallel·short parallel(A) = maxshort parallelxshort parallel = 1, xgreater-or-equal, slanted0short parallelA circle times operator xshort parallel and image.
