In this talk, some nonlinear mappings having periodic behavior, which is in the sense as generating sequences of identically periodic for all choices of initial data, will be analyzed in the way of spectrum resolution on matrix representation.Especially, we will investigate some concrete examples of piecewise-linear periodic mappings together with the rational mappings related through the ultradiscrete limit.We will see that the matrix represen tation of such mapping is still preserved under the ultradiscretization and the spectrum resolution gives us more detailed information than the period.