Aplicações de contração-dilatação para tensores

Abstract

In this dissertation, we present a study on tensors, the tensor product and its fundamental properties. Starting from a linearly ordered set of indices, we define the spaces indexed by this set and construct an algebra. In this context, we introduce the stretching maps, which have notable properties, such as the fact that they are homomorphisms, in addition to standing out as a tool for representing tensors in matrix form. We demonstrate that, under specific conditions, the tensor product of square matrices can be interpreted as a particular case of stretching maps. Additionally, we characterize injective stretching maps up to similarity and, by presenting the averaging operator, we show that any stretching map can be expressed as the composition of the averaging operator with an injective stretching map .