A parameterization of minimal stochastic realizations

Given a rational spectral density and a minimal spectral factor, a parameterization of all minimal spectral factors is derived. A minimal realization of each minimal spectral factor is also provided. This parameterization is shown to hold under the condition of nonintersection between the spectra of two matrices. This condition, first introduced in previous work by Ferrante et al. (1993), is proved here to be also necessary. Finally these results are applied to the stochastic realization problem