Basis-Free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras

Abdulkhaev, Kamron; Shirokov, D. S.

doi:10.1007/s00006-022-01232-0

Cited by 6 publications

(5 citation statements)

References 25 publications

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“…This statement was verified using computer calculations for the cases n ≤ 6 in [1]. Now we get this statement for the case of arbitrary dimension n, because it is just reformulation of Theorem 4 in other terms.…”

Section: Generalized Vieta Theorem For Arbitrary Nsupporting

confidence: 52%

“…in the case of one Jordan block. For example, for U = 5e + 1 2 (e 2 + e 12 ), we have λ 1,2 = 5 and y 1,2 = 5e ± 1 2 (e 2 + e 12 ) in the case n = p = 2, q = 0.…”

Section: The Case N =mentioning

confidence: 99%

“…The analogues of the formulas (26) for the cases n = 5, 6 are presented in [1] (see Theorem 5.1 and Section 8). These formulas also have the form of elementary symmetric polynomials, only if we ignore the operation △, and can be interpreted as generalized noncommutative Vieta's formulas.…”

Section: The Cases N ≥mentioning

confidence: 99%

“…Note that the determinant is used to calculate the inverse in geometric algebras [19,14,3,18]. In [18,1,2], the explicit formulas for the characteristic polynomial coefficients C (k) are presented in the cases n ≤ 6. The characteristic polynomial in geometric algebras is also discussed in [12] and used to solve the Sylvester and Lyapunov equations in [17,16].…”

Section: Introductionmentioning

confidence: 99%

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Noncommutative Vieta Theorem in Clifford Geometric Algebras

Shirokov¹

2023

Preprint

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In this paper, we discuss a generalization of Vieta theorem (Vieta's formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta's formulas with the ordinary Vieta's formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand -Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. We introduce the notion of a simple basis-free formula for a determinant in geometric algebra and prove that a formula of this type exists in the case of arbitrary dimension. Using this notion, we present and prove generalized Vieta theorem in geometric algebra of arbitrary dimension. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering.

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Section: Generalized Vieta Theorem For Arbitrary Nsupporting

confidence: 52%

“…in the case of one Jordan block. For example, for U = 5e + 1 2 (e 2 + e 12 ), we have λ 1,2 = 5 and y 1,2 = 5e ± 1 2 (e 2 + e 12 ) in the case n = p = 2, q = 0.…”

Section: The Case N =mentioning

confidence: 99%

Section: The Cases N ≥mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Noncommutative Vieta Theorem in Clifford Geometric Algebras

Shirokov¹

2023

Preprint

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“…The second part is perfectly represented by a paper of Sira-Ramírez et al 23 where Lyapunov stability theory for smooth nonlinear autonomous dynamical systems is presented in terms of GA, or the paper by Kamron Abdulkhaev et al 24 where basisfree formulae for characteristic polynomial coefficients are derived in terms of GA. Structural mathematical concepts, such as the generalization of Lipschitz and spin groups, or twisted algebras of GA are targeted by Ekaterina Filimoshina et al in 25 and Masaki Matsuno in 26 , respectively.…”

Section: Ga For Mathematics and Computationmentioning

confidence: 99%

Survey of New Applications of Geometric Algebra

Hitzer¹,

Kamarianakis

Papagiannakis

et al. 2023

Preprint

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This survey introduces 101 new publications on applications of Clifford’s geometric algebras (GA) newly published during 2022 (until mid-January 2023). The selection of papers is based on a comprehensive search with Dimensions.ai, followed by detailed screening and clustering. Readers will learn about the use of GA for mathematics, computation, surface representations, geometry, image- and signal processing, computing and software, quantum computing, data processing, neural networks, medical science, physics, electric engineering, control and robotics.

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Survey of new applications of geometric algebra

Hitzer,

Kamarianakis,

Papagiannakis

et al. 2023

Math Methods in App Sciences

View full text Add to dashboard Cite

This survey introduces 101 new publications on applications of Clifford's geometric algebras (GAs) newly published during 2022 (until mid‐January 2023). The selection of papers is based on a comprehensive search with Dimensions.ai, followed by detailed screening and clustering. Readers will learn about the use of GA for mathematics, computation, surface representations, geometry, image, and signal processing, computing and software, quantum computing, data processing, neural networks, medical science, physics, electric engineering, control and robotics.

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Basis-Free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras

Cited by 6 publications

References 25 publications

Noncommutative Vieta Theorem in Clifford Geometric Algebras

Noncommutative Vieta Theorem in Clifford Geometric Algebras

Survey of New Applications of Geometric Algebra

Survey of new applications of geometric algebra

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