N-Dimensional Commutative Hypercomplex Numbers

Abstract

As summarized in the preface, hypercomplex numbers were introduced before the linear algebra of matrices and vectors. In this chapter, in which we follow a classical approach, their theory is developed mainly by means of elementary algebra, and their reference to a representation with matrices, vectors or tensors is just for the practical convenience of referring to a widely known language. In particular, the down or up position of the indexes, which in tensor calculus are named covariance and contravariance, respectively, indicates if the corresponding quantities (vectors) are transformed by a direct or inverse matrix (see Section 2.1.4). We also use Einstein’s convention for tensor calculus and omit the sum symbol on the same covariant and contravariant indexes; in particular, we indicate with Roman letters the indexes running from 1 to N − 1, and with Greek letters the indexes running from 0 to N − 1.