What are the most obvious generalizations of Berlekamp Massey algorithm  to matrix sequences?
 Massey, J. L., "Shift-register synthesis and BCH decoding", IEEE Trans. Information Theory IT-15 (1): 122–127
You might find this dissertation interesting. The author has collected many references of the Matrix versions of the Berlekamp-Massey algorithm (citing Rissansen, Dickinson, Thome and other papers) and its applications.