A lower bound on the dimension of minimal positive realizations for discrete time systems

In some applications, one is interested in having a state–space realization with nonnegative matrices (positive realization) of a given transfer function and it is known that such a realization may have a dimension strictly larger than the order of the transfer function itself. The aim of this letter is to provide a lower bound on the minimum dimension of a positive realization taking into account some spectral properties of nonnegative matrices