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

Closed form expressions to calculate the exponential of a general multivector (MV) in Clifford geometric algebras (GAs) Clp;q are presented for n = p + q = 3. The obtained exponential formulas were applied to find exact GA trigonometric and hyperbolic functions of MV argument. We have verified that the presented exact formulas are in accord with series expansion of MV hyperbolic and trigonometric functions. The exponentials may be applied to solve GA differential equations, in signal and image processing, automatic control and robotics.

show abstract

“…Let Det(A) be the determinant of MV [2,7,17,23]. The determinant of the sum of vector a and bivector A parts of A simplifies to…”

Section: Special Cases Of Theoremmentioning

confidence: 99%

“…However, as a first step in deriving exact formula for tanh A, at first, one must compute the exact inverse of hyperbolic cosine. How to compute the inverse MV in case of general Clifford algebras is described in [2,14,23].…”

Section: Also Trigonometric and Hyperbolic Functions Of MVmentioning

confidence: 99%

Exponentials of general multivector in 3D Clifford algebras

Dargys

Acus

2022

NAMC

View full text Add to dashboard Cite

show abstract

“…Let Det(A) be the determinant of MV [9,10,11,12]. The determinant of the sum of vector a and bivector A parts of A simplifies to…”

Section: Exponential Inmentioning

confidence: 99%

“…However, as a first step in deriving exact formula for tanh A at first one must compute the exact inverse of hyperbolic cosine. How to compute the inverse MV in case of general Clifford algebras is described in [11,12,15]. For this purpose the adjoint and determinant of MV may be needed,…”

Section: Exponent Of Scalar + Pseudoscalarmentioning

confidence: 99%

Square root of a multivector in 3D Clifford algebras

Dargys

Acus

2020

NAMC

View full text Add to dashboard Cite

The problem of square root of multivector (MV) in real 3D (n = 3) Clifford algebras Cl 3,0, Cl 2,1, Cl 1,2 and Cl 0,3 is considered. It is shown that the square root of general 3D MV can be extracted in radicals. Also, the article presents basis-free roots of MV grades such as scalars, vectors, bivectors, pseudoscalars and their combinations, which may be useful in applied Clifford algebras. It is shown that in mentioned Clifford algebras, there appear isolated square roots and continuum of roots on hypersurfaces (infinitely many roots). Possible numerical methods to extract square root from the MV are discussed too. As an illustration, the Riccati equation formulated in terms of Clifford algebra is solved.The Clifford algebra Cl p,q is an associative noncommutative algebra, where (p, q) indicates vector space metric. In 3D case, the MV consists of the following elements (basis blades) {1, e 1 , e 2 , e 3 , e 12 , e 13 , e 23 , e 123 ≡ I}, where e i are the orthogonal basis vectors, and e ij are the bivectors (oriented planes). The last term is called the pseudoscalar. The shorthand notation, for example, e ij means e ij = e i • e j , where the circle indicates the geometric or Clifford product. Usually, the multiplication symbol is omitted, and one writes e ij = e i e j . The number of subscripts indicates the grade of basis element, so that the scalar is a grade-0 element, the vectors are the grade-1 elements etc. In the orthonormalized basis, the products of vectors satisfies e i e j + e j e i = ±2δ ij .

show abstract

“…In Section 2, we present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. In Section 3, we discuss the notion of characteristic polynomial in geometric algebras and remind the recursive formulas for characteristic polynomial coefficients from [25]. In Section 4, we present an analytic proof of the basis-free formulas for characteristic polynomial coefficients in the case n = 4.…”

Section: Introductionmentioning

confidence: 99%

Basis-free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras

Abdulkhaev,

Shirokov

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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 are verified using computer calculations. We present an analytical proof of all formulas in the case n = 4, and one of the formulas in the case n = 5. We present some new properties of the operations of conjugation and grade projection and use them to obtain the results of this paper. We also present formulas for characteristic polynomial coefficients in some special cases. In particular, the formulas for vectors (elements of grade 1) and basis elements are presented in the case of arbitrary n, the formulas for rotors (elements of spin groups) are presented in the cases n ≤ 5. The results of this paper can be used in different applications of geometric algebras in computer graphics, computer vision, engineering, and physics. The presented basis-free formulas for characteristic polynomial coefficients can also be used in symbolic computation.

show abstract

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

Cited by 24 publications

References 17 publications

Exponentials of general multivector in 3D Clifford algebras

Exponentials of general multivector in 3D Clifford algebras

Square root of a multivector in 3D Clifford algebras

Basis-free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras

Contact Info

Product

Resources

About