Square root of a multivector in 3D Clifford algebras

Abstract

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 .

Square root of a multivector of Clifford algebras in 3D: A game with signs

“…In this paper we continue and elaborate our investigations [13] of the problem of the square root in real CAs for n = 3 case. In particular, we provide explicit conditions when an arbitrary MV has the square root and how to express it in radicals.…”

“…The denominator in S 1,2 becomes zero if s = S = 0, i.e., in the case 3). The remaining two solutions of (13), which can be obtained from ( 16) by the substitution √ T 2 + t 2 → − √ T 2 + t 2 , are complex valued due to the inequality √ T 2 + t 2 ≥ T and therefore must be rejected. The two real-valued solutions of Eq.…”

“…Our preliminary investigation [13] on this subject was concerned with square roots of individual MV grades such as scalar, vector, bivector, pseudoscalar, or their simple combinations. For this purpose we have applied the Gröbner basis algorithm to analyze the system of nonlinear polynomial equations that ensue from the MV equation A 2 = B, where A and B are the MVs.…”

Exponentials of general multivector in 3D Clifford algebras

“…Since the formulas for a + and a − are expressed through square roots, it is interesting to find a MV to which the square roots are associated. In [3,11] an algorithm to compute the square root of MV in 3D algebras is provided. It seems reasonable to conjecture that the special cases in exponential are related to isolated square roots of the center a S + a I I of the considered algebra, where the scalars a S and a I are defined by…”

“…We have found that the optimal factor must be larger than the determinant norm of MV in Eq. (11). The norm is defined as the determinant of A raised to fractional power 1/k, where k = 2 n/2 , i.e., |A| = (Det(A)) 1/k .…”