On the rational generating functions of Toeplitz matrices

In the present paper, the authors provide complete answers to these two questions in the case where T is a nonzero Toeplitz matrix by introducing the characteristic degrees and the characteristic polynomial pair of T. It is important to emphasize that the characteristic degrees defined here are different from the characteristics of a nonzero Toeplitz matrix as defined and used by Heinig and Rost. At the end of the paper, the authors present two examples to demonstrate the feasibility and efficiency of their proposed method.