Dzhaliuk N. Factorization of matrices over polynomial and similar to them rings

The thesis is devoted to the investigation of factorizations of matrices over the polynomial rings, principal ideal domains and some finitely generated principal ideal rings and the developing of the methods of solution of linear and polynomial matrix equations. The monic divisors and common monic divisors of the polynomial matrices over arbitrary field with the given canonical diagonal forms are described. These divisors are found under the conditions of parallelism of the corresponding factorizations of matrices to the factorizations of their canonical diagonal forms. The linear monic triangular divisors with the one elementary divisor and of the simple structure of the polynomial matrices over an algebraic closed fields are described. The criterion of uniqueness of the solutions of matrix equations of the Sylvester type over principal ideal domains and some finitely generated principal ideal rings are established. The method of constructing of the solutions of such matrix equations is suggested. The block-triangular and block-diagonal parallel factorizations of the partitioned matrices over a principal ideal domain up to the association are described. The necessary and sufficient conditions of their uniqueness are formulated.The method of constructing of such factorizations of matrices is suggested. The established results were obtained with the aid of the linear algebra and the theory of ringsmethods and could be applyed for further study of the structure of matrices over rings, for the solution of systems of differential equations, in the theory of operator pencils and in applied problems.