Shavarovskii B. Reduction and decomposition of matrices over polynomial rings

The dissertation deals with the study of the matrix structure from the viewpoint of their reduction by different transformations to simpler forms and decomposition into factors of lower degrees. In this connection a problem of reduction of rather wide classes of polynomial matrices to direct sum of summands has been solved. This enabled to reduce complex problems of semiscalar equivalence and similarity of matrices to analogous problems for matrices of lower dimension. The systems of invariants of reduced matrices have been found and for some sets of matrices the canonical forms have been constructed. For such sets the problem of classification of matrices up to semiscalar equivalence has been solved. The decomposabiliti into factors (in particular linear) of some classes of matrix polynomials with elementary divisors of degrees more than two has been proved, the classes including the majority of known now classes of factorable matrix polynomials. The results obtained have been applied to the problems of construction the canonical forms matrix polynomials relating to similarity transformation and also to solution of various types of matrix algebraic equations (polynomial higher degrees, linear two-sided, Riccati equations).