Abstract
The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation, and the operations of conjugation. To obtain the results, we use the concepts of characteristic polynomial, determinant, adjugate, and inverse in Clifford algebras. For the first time, we give alternative formulas for the basis-free solution to the Sylvester equation in the case \(n=4\), the proofs for the case \(n=5\) and the case of arbitrary dimension n. The results can be used in symbolic computation.
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Notes
The definitions of adjugate \(\mathrm{Adj}(D)\), determinant \(\mathrm{Det}(D)\), and inverse \(D^{-1}\) in \({C}\!\ell _{p,q}\) for an arbitrary n are given in [25].
Analytic proof is also possible using the methods from [1].
Explicit formulas for the coefficients \(b_{(1)}\), ..., \(b_{(8)}\) are presented in [1].
Note that the same is not true for the expressions for characteristic polynomial coefficients from Theorem 3.1. For example, \(B\widehat{\widetilde{B}}(\widehat{B} \widetilde{B})^{\triangle }\in {C}\!\ell ^0_{p,q}\oplus {C}\!\ell ^1_{p,q}\oplus {C}\!\ell ^4_{p,q}\ne \mathrm{cen}({C}\!\ell _{p,q})\) in the case \(n=5\), see the details in [25].
Analytic proof is also possible using the methods from [1].
Here and below we denote the integer part of the number \(\frac{n+1}{2}\) by \([\frac{n+1}{2}]\).
Note that using the recursive formulas \(B_{(k+1)}=B (B_{(k)}-b_{(k)})\), the expression (5.5) can be reduced to the form \(\sum _{i, j} b_{ij} A^i C B^j\) with some scalars \(b_{ij}\in {\mathbb R}\).
In the case of odd n, the integer part of the number \(\frac{n+1}{2}\) is equal to \([\frac{n+1}{2}]=\frac{n+1}{2}\in {{\mathbb {Z}}}\).
Here and below we denote the integer part of \(\log _2 n\) by \([\log _2 n\)].
In the above example, we use the three operations \(\widehat{B}\), \(\widetilde{B}\), and \(B^\triangle \) in the cases \(4 \le n \le 7\), the four operations \(\widehat{B}\), \(\widetilde{B}\), \(B^\triangle \), and \(B^\square \) in the cases \(8 \le n \le 15\).
Note that \(B^{\triangle _1}=\widehat{B}\), \(B^{\triangle _2}=\widetilde{B}\), \(B^{\triangle _3}=B^\triangle \), and \(B^{\triangle _4}=B^\square \).
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Acknowledgements
This work is supported by the grant of the President of the Russian Federation (project MK-404.2020.1). The research presented in this paper was stimulated by discussions of the author with Prof. Hongbo Li during scientific visit to the Chinese Academy of Sciences, Academy of Mathematics and Systems Science (Beijing, China) in 2019, for which the author is grateful. The author is grateful to Prof. Nikolay Marchuk for fruitful discussions. The results of this paper were reported at the International Conference “Computer Graphics International 2020 (CGI2020)” (Geneva, Switzerland, October 2020). The author is grateful to the organizers and the participants of this conference for fruitful discussions. The author is grateful to the editor, Prof. Eckhard Hitzer, and two anonymous reviewers for their careful reading of the paper and helpful comments on how to improve the presentation.
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This article is part of the ENGAGE 2020 Topical Collection on Geometric Algebra for Computing, Graphics and Engineering edited by Werner Benger, Dietmar Hildenbrand, Eckhard Hitzer, and George Papagiannakis.
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Shirokov, D. Basis-free Solution to Sylvester Equation in Clifford Algebra of Arbitrary Dimension. Adv. Appl. Clifford Algebras 31, 70 (2021). https://doi.org/10.1007/s00006-021-01173-0
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DOI: https://doi.org/10.1007/s00006-021-01173-0
Keywords
- Clifford algebra
- Geometric algebra
- Sylvester equation
- Lyapunov equation
- characteristic polynomial
- Basis-free solution