Abstract
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 the operations of multiplication, summation, and operations of conjugation without explicit use of matrix representation. We use methods of Clifford algebras (including the method of quaternion typification proposed by the author in previous papers and the method of operations of conjugation of special type presented in this paper) and generalizations of numerical methods of matrix theory (the Faddeev–LeVerrier algorithm based on the Cayley–Hamilton theorem; the method of calculating the characteristic polynomial coefficients using Bell polynomials) to the case of Clifford algebras in this paper. We present the construction of operations of conjugation of special type and study relations between these operations and the projection operations onto fixed subspaces of Clifford algebras. We use this construction in the analytical proof of formulas for the determinant, other characteristic polynomial coefficients, adjugate, and inverse in Clifford algebras. The basis-free formulas for the inverse give us basis-free solutions to linear algebraic equations, which are widely used in computer science, image and signal processing, physics, engineering, control theory, etc. The results of this paper can be used in symbolic computation.
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Notes
One of the anonymous reviewers noted that he implemented and tested (both recursive and explicit) algorithm and checked that it yields correct results for Clifford algebras up to dimension \(n=11\) (using elements with random integer coefficients), and stated that explicit algorithm is much more efficient than recursive; he also computed (using optimized version of the formula, provided by Lemma 6 in the article) of symbolic expression for determinant for \({\mathcal {C}}\ell _{6,0}\) in expanded form in approximately 1 day, whereas symbolic matrix determinant computation (in expanded form) took more than 4 days.
See also applications of the results of this paper in symbolic computation using the Mathematica package (Acus and Dargys 2017), https://github.com/ArturasAcus/GeometricAlgebra, and the Python package (Arsenovic et al. 2018), https://github.com/pygae/clifford/pull/373.
For example, the subspace \({\mathcal {C}}\ell ^{\overline{0}}_{p,q}\) consists of elements that are not changing under the grade involution \(\widehat{U}=U\) and the reversion \(\widetilde{U}=U\), i.e., elements of grades 0, 4, 8, etc. The subspace \({\mathcal {C}}\ell ^{\overline{1}}_{p,q}\) consists of elements that satisfy \(\widehat{U}=-U\) and \(\widetilde{U}=U\), i.e., elements of grades 1, 5, 9, etc. Similarly for the other two subspaces. Other properties of these four subspaces are discussed in detail in Shirokov (2012a, 2012b).
As one of the anonymous reviewers correctly noted, these formulas can also be used as definitions of projection operations onto the subspaces of quaternion types 0, 1, 2, and 3.
The equivalence of these two definitions follows from the following fact: the binomial coefficient \(C^i_k\) is odd if and only if there are no 1 in the binary notation of the number i in the digits, in which there is 0 in the binary notation of the number k.
It can be proved that all operations \(\langle U \rangle _k\), \(k=1, \ldots , n\) (similarly to the case of the operation \(\langle U\rangle _0\), see Theorem 1) can be realized as linear combinations of the operations \(\mathrm{id}\), \(\vartriangle _1\), \(\ldots \), \(\vartriangle _m\), \(\vartriangle _1 \vartriangle _2\), \(\ldots \), \(\vartriangle _1\cdots \vartriangle _m\). As a consequence, we get that all operations of conjugation (7) can be realized as linear combinations of the operations \(\mathrm{id}\), \(\vartriangle _1\), \(\ldots \), \(\vartriangle _m\), \(\vartriangle _1 \vartriangle _2\), \(\ldots \), \(\vartriangle _1\cdots \vartriangle _m\). We do not use this fact in this paper.
Note that some authors (Lounesto 2001) denote by
the operation of Clifford conjugation. We denote the Clifford conjugation by two symbols \(\,\widetilde{\widehat{\,}}\,\) in this paper so that there is no confusion.And more than 400 other formulas obtained from the presented here two formulas: we can take the reversion, the grade involution, or the Clifford conjugation of the scalar N(U); we can do cyclic permutations of multipliers in the obtained products because the left inverse equals to the right inverse; we can use the properties of the operation \(\vartriangle \) (28) and (29); also, we can use \(N(U)=N(\widehat{U})=N(\widetilde{U})\) because of the results of Sect. 4. We do not present all these formulas here because of their large number.
And other formulas for F(U) in the case \(n=5\) because of the large number of different formulas for N(U), see above.
Let us give the alternative proof: \(U \widetilde{U} \widehat{U} \widehat{\widetilde{U}}=H\widehat{H}=(H_0+H_1)(H_0-H_1)= (H_0)^2+[ H_0, H_1]-(H_1)^2=(H_0)^2-(H_1)^2\in {\mathcal {C}}\ell ^0_{p,q}\). One further alternative proof: \(U \widehat{\widetilde{U}} \widehat{U} \widetilde{U}=J \widetilde{J}= (J_0+ J_3)( J_0-J_3)= (J_0)^2+[J_0, J_3]-(J_3)^2= (J_0)^2- (J_3)^2\in {\mathcal {C}}\ell ^0_{p,q}.\)
In this paper, we denote the grade-negation operations by \(\underline{k}\) (not by \(\overline{k}\) as in Acus and Dargys 2018) to avoid confusion with the notation of quaternion types.
Note that if we will use the matrix representation \(\gamma \) (47) instead of the matrix representation \(\beta \) (49) in the definition of the determinant, then we obtain another concept of the determinant with values in \({\mathbb {R}}\), \({\mathbb {C}}\), or \({\mathbb {H}}\), which does not coincide with the first one in the general case. We need not this concept in this paper but use it, for example, in Shirokov (2018). In the cases \(p-q=0, 1, 2 {\, \mathrm{mod} \,}8\), we can use the representation \(\gamma \) (47) and some fixed representation \(\gamma ^\prime \) (see the recursive method in Shirokov (2018) or the method using idempotents and basis of the left ideal in Abłamowicz 1998) instead of \(\beta \) and \(\beta ^\prime \) and obtain the same concept of the determinant.
We use the notation with indices in round brackets “(k)” to avoid confusion with the notation of the projection operations onto subspaces of fixed grades.
the operation of Clifford conjugation. We denote the Clifford conjugation by two symbols References
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Acknowledgements
The author is grateful to N. Marchuk and N. Khlyustova for useful discussions. The author is grateful to the anonymous reviewers for their careful reading of the paper and helpful comments on how to improve the presentation. The results of this paper were reported at the 9th International Conference on Mathematical Modeling (Yakutsk, 2020), the 12th International Conference on Clifford Algebras and Their Applications in Mathematical Physics (Hefei, 2020), and the International Conference “Computer Graphics International” (Geneva, 2020, within the workshop “Empowering Novel Geometric Algebra for Graphics and Engineering”). The author is grateful to the organizers and the participants of these conferences for fruitful discussions. The publication was prepared within the framework of the Academic Fund Program at the HSE University in 2020–2021 (grant 20-01-003).
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Communicated by Jinyun Yuan.
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Shirokov, D.S. On computing the determinant, other characteristic polynomial coefficients, and inverse in Clifford algebras of arbitrary dimension. Comp. Appl. Math. 40, 173 (2021). https://doi.org/10.1007/s40314-021-01536-0
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DOI: https://doi.org/10.1007/s40314-021-01536-0