Factorization of symmetric matrices over polynomial rings with involution

Geographically Separating Sectors in Multi-Objective Location-RoutingProblems

“…In the present parer we found necessary and sufficient conditions for the existence of the factorization (1) of a symmetric invertible matrix Proof. Necessity follows from Theorem 1 [10].…”

“…In this paper we present a simple method of finding the factorization of an invertible symmetric matrices over a polynomial ring uses the notions of dual polynomial matrix [3], generalized dual polynomial matrix [6] and the factorizations of such matrices. Our results are presented in recent papers [6,10,11] and they base on the theory of the factorizations of a regular symmetric matrices. In particular, in [5,6,10] using the notions of infinite elementary divisors, dual polynomial matrix and generalized dual polynomial matrix were found conditions of existence factorization of a singular matrix polynomials.…”

“…Our results are presented in recent papers [6,10,11] and they base on the theory of the factorizations of a regular symmetric matrices. In particular, in [5,6,10] using the notions of infinite elementary divisors, dual polynomial matrix and generalized dual polynomial matrix were found conditions of existence factorization of a singular matrix polynomials. In the [11] was investigated the problem of factorization of a regular symmetric matrices over polynomial rings with involution.…”

“…One of the most important problems in the matrix theory is the problem of factorization of symmetric matrices over a polynomial rings with involution, the problem of classification and investigation of the structure of their elementary divisors [7,9,10,11]. It is known that algebraic problems of those type arise in the theory of differential equations, the nonlinear control theory, the optimal control and integrated systems of classical mechanics, and so have important practical value [1,8].…”

Triangular form of Laurent polynomial matrices and their factorization

“…Due to the introduced notion of the value of the matrix on the system of roots of diagonal elements in [12] the process of establishing the conditions of regularization of the matrix polynomial and factorization of symmetric matrix polynomials is significantly simplified [13].…”