On the spectral and combinatorial structure of of 2D positive systems

The dynamics of a 2D positive system depends on the pair of nonnegative square matrices that provide the updating of its local states. In this paper, several spectral properties, such as finite memory, separability, and property L, which depend on the characteristic polynomial of the pair, are investigated under the nonnegativity constraint and in connection with the combinatorial structure of the matrices. Some aspects of the Perron-Frobenius theory are extended to the 2D case; in particular, conditions are provided guaranteeing the existence of a common maximal eigenvector for two nonnegative matrices with irreducible sum. Finally, some results on 2D positive realizations are presented.