Algorithm for finding the inverse elements and solution of the Silvester equation in the Clifford algebras R4,0, R1,3, R5,0

DOI: 10.47026/1810-1909-2023-4-109-119

УДК 517.548

ББК В161.55

Sergey P. KUZNETSOV, Vladimir V. MOCHALOV, Vasiliy P. CHUEV

Key words

real Clifford algebra, conjugation in Clifford algebras, zero divisors, inverse element, Sylvester equation, robustness, piezodrive

Abstract

The purpose of the work is to find an algorithm for finding inverse elements in the Clifford algebras R_4,0, R_1,3, R_5,0 and to solve the nonlinear Sylvester equation .

Materials and methods. Using the basic conjugation operations in Clifford algebras, finding an algorithm for finding inverse elements. Application of this algorithm to solve the Sylvester equation.

Results of the work. In Clifford algebras R_4,0, R_1,3, R_5,0, which have a great application in physics, a method for finding inverse elements and equations for finding zero divisors were found. The found algorithm is used to solve the Sylvester equation. For Clifford algebras of even dimension R_4,0, R_1,3 an algorithm for finding inverse elements is given. Finding inverse elements is closely related to the concept of zero divisors in these algebras. The inverse element method is used to solve the Sylvester equation, using even conjugation, reverse conjugation and Clifford conjugation. For the odd Clifford algebra R_5,0, a conjugation is found that can be used to apply the algorithm for finding the inverse element. The method of finding the inverse element is used to solve the Sylvester equation, which, in particular, is used to ensure the robustness of the piezodrive using the controlled relative interval method.

Findings. An algorithm for finding inverse elements is constructed and the Sylvester equation is solved in the Clifford algebras R_4,0, R_1,3, R_5,0.

References

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Information about the authors

Sergey P. Kuznetsov – Senior Lecturer, Department of Discrete Mathematics and Computer Science, Chuvash State University, Russia, Cheboksary (chevchenka@mail.ru; ORCID: https://orcid.org/0000-0002-6491-2223).

Vladimir V. Mochalov – Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Discrete Mathematics and Computer Science, Chuvash State University, Russia, Cheboksary (m622573@gmail.com; ORCID: https://orcid.org/0000-0002-8136-3932).

Vasiliy P. Chuev – Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Discrete Mathematics and Computer Science, Chuvash State University, Russia, Cheboksary (570065@mail.ru; ORCID: https://orcid.org/0000-0002-1490-0929).

For citations

Kuznetsov S.P., Mochalov V.V., Chuev V.P. ALGORITHM FOR FINDING THE INVERSE ELEMENTS AND SOLUTION OF THE SILVESTER EQUATION IN THE CLIFFORD ALGEBRAS R_4,0, R_1,3, R_5,0. Vestnik Chuvashskogo universiteta, 2023, no. 4, pp. 109–119. DOI: 10.47026/1810-1909-2023-4-109-119 (in Russian).

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