REGULARIZATION AND FACTORIZATION OF LAURENT POLYNOMIAL MATRICES
DOI:
https://doi.org/10.35546/kntu2078-4481.2023.1.4Keywords:
regular Laurent polynomial matrix, the upper and lower degrees of the Laurent polynomial matrix, regularization and factorization of the Laurent polynomial matrix, Smith normal form, matrix values on the system of roots of diagonal elements, matrix equationAbstract
In recent decades, the Laurent polynomial matrices and their factorizations have many potential applications in the fields of control and automatic control systems, theory of finite-state controlled systems, theory of image reproduction, and theory of data transmission devices. These matrices are used to describe the convolutional mixing process that occurs, for example, when a set of signals arrives at a sensor array over multiple paths. The study of factorizations of Laurent polynomial matrices is relevant and is used in multi-channel signal processing. Effective algebraic algorithms, which are based on elementary transformations of Laurent polynomial matrices and their factorizations, allow a complete analysis of system dynamics. Many problems in the field of digital signal processing and communication can also be transformed into algebraic problems of polynomial Laurent rings, and they can be solved using existing algebraic methods. The article considers the problem of regularization of the Laurent polynomial matrices and obtains the necessary and sufficient conditions for the regularization of such matrices. This result is used to study the factorization of polynomial matrices over the Laurent ring. A criterion for the factorization of polynomial matrices over the Laurent ring with a regular factor with a predetermined Smith form is obtained. A method of constructing matrix regularization and factorization over the Laurent ring of polynomials is proposed, and examples of matrix regularization and factorization over the Laurent ring are given.
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