B. De Schutter, "Upper bounds for the index of cyclicity of a matrix," Tech. rep. 98-32, ESAT-SISTA, K.U.Leuven, Leuven, Belgium, 16 pp., July 1999. Revised version.
We derive upper bounds for the index of cyclicity of a matrix as a function of the size of the matrix. This result can be used in the characterization of the ultimate behavior of the sequence of consecutive powers of a matrix in the max-plus algebra, which has maximum and addition as its basic operations. If the matrix is irreducible then it is well known that the ultimate behavior is cyclic. For reducible matrices the behavior is more complex, but it is also cyclic in nature. The length of the cycles corresponds to the index of cyclicity of the given matrix.