Construction of a matrix product stationary state from solutions of finite size system

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Stationary states of stochastic models, which have N states per site, in matrix product form are considered. First we give a necessary condition for the existence of a finite M-dimensional matrix product state for any N,M. Second, we give a method to construct the matrices from the stationary states of small size system when the above condition and N≤ M are satisfied. Third, the method by which one can check that the obtained matrices are valid for any system size is presented for the case where M=N is satisfied. The application of our methods is explained using three examples: the asymmetric exclusion process, a model studied in [F. H. Jafarpour: J. Phys. A: Math. Gen. 36 (2003) 7497] and a hybrid of both of the models.