Clifford's Algebra and Applications

In the literature, we come across well-established concepts and sufficiently rich results concerning topics such as the rational approximation of real functions to a real variable, real orthogonal polynomials to a real variable, the acceleration of convergence of a real sequence, and so on. When we need to extend these same concepts to the vector case, we are confronted with the inadequacy of the algebraic structures of a vector space. So, for example, an empirical construction of an "inverse" of a non-zero vector $x\in {ℝ}^d$ was given by the formula $\frac{x}{{\Vert x\Vert }_2^2}$ and has been used in many generalizations. Clearly, this "inverse" has no algebraic meaning, given the absence, in a vector space, of an internal multiplicative law and consequently of a neutral element. This "inverse" is referred to in the literature as the Samelson inverse or the Moore-Penrose pseudo-inverse, to emphasize the deficiency of the existence of the inverse of a vector in the algebraic sense.

The introduction of universal Clifford algebra associated with a real vector space ${ℝ}^d$ provided with a non-degenerate symmetric bilinear form, was motivated by the need to construct an internal multiplicative law such that the new structure (Clifford algebra) is an algebra, containing the space ${ℝ}^d$ . The algebra is associative, non-commutative. The body $ℂ$ the body $ℍ$ quaternions are simple early examples of Clifford algebra. Although it is not integral in general, any non-zero vector $x$ from ${ℝ}^d$ considered as a Euclidean vector space, has one inverse and only one in the algebraic sense, in the Clifford algebra associated with ${ℝ}^d$...