The eigenvectors of a fuzzy matrix correspond to steady states of a complex discrete-events system, characterized by the given transition matrix and fuzzy state vectors. The descriptions of the eigenspace (the set of all eigenvectors) for matrices in the max-min, max-Łukasiewicz or max-drast fuzzy algebra have been presented in previous papers. The eigenspace of a fuzzy matrix in the max-prod algebra is investigated in this paper. First, necessary and sufficient conditions are shown under which the eigenspace restricted to increasing eigenvectors of a given matrix is non-empty, and the structure of the increasing eigenspace is described. Then, using simultaneous row and column permutations of the matrix, the complete characterization of the whole eigenspace structure of a given fuzzy matrix is shown. The details for matrices of order 3 are only presented. The method works analogously for square matrices of higher orders, with rapidly increasing complexity of the formulas.