Kuchma M. Factorizations of symmetric matrices over rings of polynomials with involution.

The dissertation is devoted to investigating the factorization of symmetric matrices over rings of polynomials with involution. Criterions of factorization of symmetric matrices over rings of polynomials and quasipolynomials with involution are obtained. The class of matrices, matrix binomials which are always factorized, is described. Considering notions of invertible and generalized invertible matrix polynomials and applying infinite elementary divisors, there were established necessary and sufficient conditions of separation from singular polynomial matrix singular factor with Smith form and system of infinite elementary divisors. Conditions on existence of factorization of singular symmetric matrices over rings of polynomials with involution are found. The connection between factorizations of regular and singular polynomial matrices over rings with involution are described. The notion of common factorization and symmetric equivalence is introduced and conditions on existence of such factorization a re obtained. Key words: factorization, regular (singular) matrix polynomial, common factorization, symmetric equivalence, Smith form.