Petrychkovych V. Generalized equivalence of matrices and its collections and factorization of matrices over rings

The thesis devoted to study the equivalences of finite collections of matrices over rings as well as to factorization of matrices. It is established the standard form for a pair of matrices over an adequate ring with respect to introduced concept of generalized equivalence. The conditions of generalized equivalence of pairs of matrices and a criterion of theirs diagonabilisity are pointed out. It is elaborated a method of factorization of matrices. The parallel factorization of matrices over adequate rings and rings of polynomials over an arbitrary field are described. It is pointed out the criterions of uniqueness of factorizations of matrices and theirs divisors. It is established the conditions of decomposability of a polynomial matrix into monic factors depending on the multiplicity of its characteristic roots and of degrees of elementary divisors. The bounds for the number of linear monic divisors of a polynomial matrix are found and the relationships between the number of divisors and the reducibility of matrices are pointed out.