On computing the determinant, other characteristic polynomial coefficients, and inverse in Clifford algebras of arbitrary dimension

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.

arXiv comment24 pages

References in corpus (9)

• Clifford algebra, geometric algebra, and applications
• Quaternion typification of Clifford algebra elements
• A classification of Lie algebras of pseudounitary groups in the techniques of Clifford algebras
• Development of the method of quaternion typification of Clifford algebra elements
• Classification of Lie algebras of specific type in complexified Clifford algebras
• Inverse and Determinant in 0 to 5 Dimensional Clifford Algebra
• Inverse of multivector: Beyond p+q=5 threshold
• Notions of determinant, spectrum, and Hermitian conjugation of Clifford algebra elements
• A note on solution of $ax+xb=c$ by Clifford algebras

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• Machine Learning Clifford invariants of ADE Coxeter elements
• Development of the Method of Averaging in Clifford Geometric Algebras
• On Rank of Multivectors in Geometric Algebras
• Determinant, Characteristic Polynomial, and Inverse in Commutative Analogues of Clifford Algebras
• Introducing Multidimensional Dirac-Hestenes Equation
• On Unitary Groups in Ternary and Generalized Clifford Algebras
• On Commutative Analogues of Clifford Algebras and Their Decompositions