60 research outputs found
Multivector and multivector matrix inverses in real Clifford algebras
We show how to compute the inverse of multivectors in finite dimensional real Clifford algebras Cl(p, q). For algebras over vector spaces of fewer than six dimensions, we provide explicit formulae for discriminating between divisors of zero and invertible multivectors, and for the computation of the inverse of a general invertible multivector. For algebras over vector spaces of dimension six or higher, we use isomorphisms between algebras, and between multivectors and matrix representations with multivector elements in Clifford algebras of lower dimension. Towards this end we provide explicit details of how to compute several forms of isomorphism that are essential to invert multivectors in arbitrarily chosen algebras. We also discuss briefly the computation of the inverses of matrices of multivectors by adapting an existing textbook algorithm for matrices to the multivector setting, using the previous results to compute the required inverses of individual multivectors
Multivector and multivector matrix inverses in real Clifford algebras
We show how to compute the inverse of multivectors in finite dimensional real Clifford algebras Cl(p, q). For algebras over vector spaces of fewer than six dimensions, we provide explicit formulae for discriminating between divisors of zero and invertible multivectors, and for the computation of the inverse of a general invertible multivector. For algebras over vector spaces of dimension six or higher, we use isomorphisms between algebras, and between multivectors and matrix representations with multivector elements in Clifford algebras of lower dimension. Towards this end we provide explicit details of how to compute several forms of isomorphism that are essential to invert multivectors in arbitrarily chosen algebras. We also discuss briefly the computation of the inverses of matrices of multivectors by adapting an existing textbook algorithm for matrices to the multivector setting, using the previous results to compute the required inverses of individual multivectors
Construction of multivector inverse for Clifford algebras over 2m+1-dimensional vector spaces from multivector inverse for Clifford algebras over 2m-dimensional vector spaces
Assuming known algebraic expressions for multivector inverses in any Clifford algebra over an even dimensional vector space Rp′,q′, n ′ = p ′ + q ′ = 2 m, we derive a closed algebraic expression for the multivector inverse over vector spaces one dimension higher, namely over R p,q , n= p+ q= p ′ + q ′ + 1 = 2 m+ 1. Explicit examples are provided for dimensions n ′ = 2 , 4 , 6 , and the resulting inverses for n= n ′ + 1 = 3 , 5 , 7. The general result for n= 7 appears to be the first ever reported closed algebraic expression for a multivector inverse in Clifford algebras Cl(p, q), n= p+ q= 7 , only involving a single addition of multivector products in forming the determinant
Algorithmic computation of multivector inverses and characteristic polynomials in non-degenerate Clifford algebras
The power of Clifford or, geometric, algebra lies in its ability to represent
geometric operations in a concise and elegant manner. Clifford algebras provide
the natural generalizations of complex, dual numbers and quaternions into
non-commutative multivectors. The paper demonstrates an algorithm for the
computation of inverses of such numbers in a non-degenerate Clifford algebra of
an arbitrary dimension. The algorithm is a variation of the
Faddeev-LeVerrier-Souriau algorithm and is implemented in the open-source
Computer Algebra System Maxima. Symbolic and numerical examples in different
Clifford algebras are presented.Comment: 11 pages. arXiv admin note: text overlap with arXiv:1904.0008
Clifford Multivector Toolbox (for MATLAB)
matlab ® is a numerical computing environment oriented towards manipulation of matrices and vectors (in the linear algebra sense, that is arrays of numbers). Until now, there was no comprehensive toolbox (software library) for matlab to compute with Clifford algebras and matrices of multivectors. We present in the paper an account of such a toolbox, which has been developed since 2013, and released publically for the first time in 2015. The paper describes the major design decisions made in implementing the toolbox, gives implementation details, and demonstrates some of its capabilities, up to and including the LU decomposition of a matrix of Clifford multivectors
On computing the determinant, other characteristic polynomial coefficients, and inverse in Clifford algebras of arbitrary dimension
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 . 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.Comment: 24 page
- …