# 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.

## Keywords

Characteristic Matrix Canonical System Invariant Quantity Unity Versor Characteristic Determinant## Preview

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