Products of tensors

Abstract

This dissertation surveys the products of tensors, their respective multilinear structures and their properties. Matrix products arise naturally from the composition of linear operations. In like manner, products of tensors derive from composing multilinear functions. However, there are many ways to compose multilinear functions, hence many forms of the product. We will gain further insight of linear operations and introduce the tensor-matrix multiplication (the simplest form of the compositional product). We will consider different types of products building up to the study of the general tensor-tensor multiplication defined in [Lin. Alg. Appl. 439, 2350–2366 (2013)]. Finally, we conclude with an investigation of some n-ary products of tensors.