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## On Noncommutative Vieta Theorem in Geometric Algebras

P. 28–37.

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. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering.

Dmitry Shirokov, Mathematical Methods in the Applied Sciences 2024 Vol. 47 No. 14 P. 11305–11320

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 formulas with the ordinary Vieta 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 ...

Added: April 2, 2023

Kamron Abdulkhaev, Shirokov D., Advances in Applied Clifford Algebras 2022 Vol. 32 No. 5 Article 57

In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras Gp,q of vector space of dimension 𝑛=𝑝+𝑞. We present basis-free formulas for all characteristic polynomial coefficients in the cases 𝑛≤6, alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations of conjugation. All the formulas ...

Added: October 11, 2022

Abdulkhaev K., Shirokov D., , in: Advances in Computer Graphics: 38th Computer Graphics International Conference, CGI 2021, Virtual Event, September 6–10, 2021, Proceedings. Springer, 2021. P. 670–681.

Added: September 19, 2021

Shirokov D., Computational and Applied Mathematics 2021 Vol. 40 P. 1–29

In this paper, we solve the problem of computing the inverse in Clifford algebras of arbitrary dimension. We present basis-free formulas of different types (explicit and recursive) for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in real Clifford algebras (or geometric algebras) over vector spaces of arbitrary dimension $n$. The formulas involve only ...

Added: July 15, 2021

Shirokov D., Advances in Applied Clifford Algebras 2021 Vol. 31 P. 1–19

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 ...

Added: September 19, 2021

Shirokov D., , in: Advances in Computer Graphics. CGI 2020. Springer, 2020. P. 541–548.

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 the basis-free solution to the Sylvester equation in geometric algebra of arbitrary dimension. The basis-free solutions involve only the operations of geometric product, summation, and the ...

Added: November 2, 2020

Shirokov D., , in: Advances in Computer Graphics: 40th Computer Graphics International Conference, CGI 2023, Shanghai, China, August 28 – September 1, 2023, Proceedings, Part IV* 4. Vol. 14498. Springer, 2024. P. 391–401.

This paper is a brief note on the natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real (Clifford) geometric algebras of arbitrary dimension and signature. We naturally define these and other related structures (operation of Hermitian conjugation, Euclidean space, and Lie groups) in geometric algebras. The results ...

Added: December 25, 2023

Shirokov D., , in: AIP Conference ProceedingsVol. 2328: ICMM-2020. AIP Publishing LLC, 2021. Ch. 060001 P. 060001-1–060001-4.

In this note, we present basis-free definitions of subspaces of fixed grades of real Clifford algebras of arbitrary dimension. We do not use fixed basis of Clifford algebra and use only the properties of commutators and anticommutators. ...

Added: April 2, 2021

Shirokov D., Advances in Applied Clifford Algebras 2019 Vol. 29 No. 50 P. 1–12

We present a method of computing elements of spin groups in the case of arbitrary dimension. This method generalizes Hestenes method for the case of dimension 4. We use the method of averaging in Clifford’s geometric algebra previously proposed by the author. We present explicit formulas for elements of spin group that correspond to the ...

Added: July 22, 2019

Shirokov D., Advances in Applied Clifford Algebras 2024 Vol. 34 Article 23

In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related ...

Added: August 23, 2024

Shirokov D., Advances in Applied Clifford Algebras 2021 Vol. 31 Article 30

In this paper, we consider inner automorphisms that leave invariant fixed subspaces of real and complex Clifford algebras — subspaces of fixed grades and subspaces determined by the reversion and the grade involution. We present groups of elements that define such inner automorphisms and study their properties. Some of these Lie groups can be interpreted ...

Added: May 10, 2021

Ekaterina Filimoshina, Dmitry Shirokov, Mathematical Methods in the Applied Sciences 2024 Vol. 47 No. 3 P. 1375–1400

This paper presents some new Lie groups preserving fixed subspaces of geometric algebras (or Clifford algebras) under the twisted adjoint representation. We consider the cases of subspaces of fixed grades and subspaces determined by the grade involution and the reversion. Some of the considered Lie groups can be interpreted as generalizations of Lipschitz groups and ...

Added: October 11, 2022

Filimoshina E., Shirokov D., , in: Advanced Computational Applications of Geometric Algebra: First International Conference, ICACGA 2022, Denver, CO, USA, October 2-5, 2022, ProceedingsVol. 13771. Springer, 2024. P. 186–198.

In this paper, we introduce and study several Lie groups in degenerate (Clifford) geometric algebras. These Lie groups preserve the even and odd subspaces under the adjoint representation and the twisted adjoint representation. The considered Lie groups are interesting for the study of spin groups and their generalizations in degenerate case. ...

Added: February 3, 2024

Rumiantseva S., Shirokov D., Advances in Applied Clifford Algebras 2024 P. 1–20

It is easier to investigate phenomena in particle physics geometrically by exploring a real solution to the Dirac-Hestenes equation instead of a complex solution to the Dirac equation. The present research outlines the multidimensional Dirac-Hestenes equation. Since the matrix representation of the complexified (Clifford) geometric algebra ℂ⊗Cℓ1,n depends on a parity of n, we explore even and odd ...

Added: November 8, 2024

Ekaterina Filimoshina, Dmitry Shirokov, Advances in Applied Clifford Algebras 2023 Vol. 33 Article 44

In this paper, we introduce and study five families of Lie groups in degenerate Clifford geometric algebras. These Lie groups preserve the even and odd subspaces and some other subspaces under the adjoint representation and the twisted adjoint representation. The considered Lie groups contain degenerate spin groups, Lipschitz groups, and Clifford groups as subgroups in ...

Added: August 19, 2023

Filimoshina E., Dmitry Shirokov, Advances in Applied Clifford Algebras 2024 Vol. 34 Article 50

This paper investigates centralizers and twisted centralizers in degenerate and non-degenerate Clifford (geometric) algebras. We provide an explicit form of the centralizers and twisted centralizers of the subspaces of fixed grades, subspaces determined by the grade involution and the reversion, and their direct sums. The results can be useful for applications of Clifford algebras in ...

Added: November 8, 2024

Shirokov D., Mathematics 2023 Vol. 11 No. 16 Article 3607

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Open AccessFeature PaperArticle
Development of the Method of Averaging in Clifford Geometric Algebras
by
Dmitry Shirokov
1,2
1
HSE University, Myasnitskaya Str. 20, Moscow 101000, Russia
2
Institute for Information Transmission Problems of Russian Academy of Sciences, Bolshoy Karetny Per. 19, Moscow 127051, Russia
Mathematics 2023, 11(16), 3607; https://doi.org/10.3390/math11163607
Received: 29 June 2023 / Revised: 15 August 2023 / Accepted: 17 August 2023 / Published: 21 August 2023
(This article belongs to the ...

Added: October 5, 2023

Dogra R., Lando S., Communications in Mathematics 2023 Vol. 31 No. 3 P. 87–111

We introduce a new one-variable polynomial invariant of graphs, which we call the skew characteristic polynomial. For an oriented simple graph, this is just the characteristic polynomial of its anti-symmetric adjacency matrix. For nonoriented simple graphs the definition is different, but for a certain class of graphs (namely, for intersection graphs of chord diagrams), it ...

Added: February 19, 2024

Shirokov D., Advances in Applied Clifford Algebras 2018 Vol. 28 No. 3 P. 1–16

We present a new class of covariantly constant solutions of the Yang–Mills equations. These solutions correspond to the solution of the field equation for the spin connection of the general form. ...

Added: July 6, 2018

Covolo T., Journal of Noncommutative Geometry 2015 Vol. 9 No. 2 P. 543–565

We develop the theory of linear algebra over a (Z2)n-commutative algebra (n∈N), which includes the well-known super linear algebra as a special case (n=1). Examples of such graded-commutative algebras are the Clifford algebras, in particular the quaternion algebra H. Following a cohomological approach, we introduce analogues of the notions of trace and determinant. Our construction ...

Added: September 28, 2015

Shirokov D., Marchuk N., Advances in Applied Clifford Algebras 2008 Vol. 18 No. 2 P. 237–254

For the complex Clifford algebra <img /> (p, q) of dimension n = p + q we define a Hermitian scalar product. This scalar product depends on the signature (p, q) of Clifford algebra. So, we arrive at unitary spaces on Clifford algebras. With the aid of Hermitian idempotents we suggest a new construction of, so called, normal matrix representations of Clifford algebra elements. These ...

Added: June 16, 2015

Shirokov D., Advances in Applied Clifford Algebras 2015 Vol. 25 No. 1 P. 227–244

We formulate generalizations of Pauli’s theorem on the cases of real and complex Clifford algebras of even and odd dimensions. We give analogues of these theorems in matrix formalism. Using these theorems we present an algorithm for computing elements of spin groups that correspond to elements of orthogonal groups as double cover. ...

Added: March 11, 2015

Shirokov D., Advances in Applied Clifford Algebras 2015 Vol. 25 No. 3 P. 707–718

In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of corresponding Lie algebras. ...

Added: March 12, 2015

Burman Y. M., / Cornell University. Series math "arxiv.org". 2012. No. arXiv:1205.1123.

We calculate characteristic polynomials of operators explicitly represented as polynomials of rank $1$ operators. Applications of the results obtained include a generalization of the Forman--Kenyon's formula for a determinant of the graph Laplacian and also provide its level $2$ analog involving summation over triangulated nodal surfaces with boundary. ...

Added: May 15, 2012