  Asymptotic properties of matrices are, in general, difficult to analyze with
classical mathematical techniques. In very specific cases, there is a
well-known connection between the asymptotic behavior of a matrix's leading
eigenvector and the corresponding "tropical" matrix, arising out of the $max$
and $min$ operations innate in tropical analysis. In this paper we examine a
more general class of matrices, and explore the extent to which we can
generalize the results using tropical techniques. We find that while the
original results do not easily generalize, we can still make some useful
statements about the asymptotic behavior in the general case, and can give a
complete characterization for a larger class of matrices than previously
examined.
