FACTORIZATION OF LAURENT SYMMETRIC POLYNOMIAL MATRICES OVER THE RING WITH INVOLUTION - Kuchma M. І.; Dilnyi V. М.; Oryshchyn O. Н. (PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, 2024)
Many problems in the fields of digital signal processing and communication, automatic control theory, theory of finite-state controlled systems, theory of image reproduction, and theory of data transmission devices can be converted into algebraic problems for Laurent polynomial matrices and can be solved using existing algebraic methods. Effective algebraic algorithms, which are based on elementary transformations of Laurent polynomial matrices and their factorizations, allow a complete analysis of system dynamics. The introduced concept of semiscalar equivalence of Laurent polynomial matrices and the established lower triangular form with invariant factors on the main diagonal for them, as well as the obtained condition for regularization of Laurent polynomial matrices, allows us to find conditions for the factorization of symmetric Laurent matrices on the ring of polynomials with involution. The article deals with the problem of factorization of symmetric Laurent polynomial matrices over a ring with involution. Necessary and sufficient conditions for the factorization of such matrices with a regular factor with a predefined canonical Smith form were obtained. A criterion for the factorization of Laurent symmetric polynomial matrices is obtained, which is parallel to the Smith factorization of its form.