General Techniques

Abstract

In this chapter, isomorphisms between the tensor space of order d and vector spaces or other tensor spaces are considered. The vectorisation from Sect. 5.1 ignores the tensor structure and treats the tensor space as a usual vector space. In finite dimensional implementations this means that multivariate arrays are organised as linear arrays. After vectorisation, linear operations between tensor spaces become matrices expressed by Kronecker products (cf. §5.1.2).

While vectorisation ignores the tensor structure completely, matricisation keeps one of the spaces and leads to a tensor space of order two (cf. Sect. 5.2). In the finite dimensional case, this space is isomorphic to a matrix space. The interpretation as matrix allows to formulate typical matrix properties like the rank leading to the j-rank for a direction j and the α-rank for a subset α of the directions 1, …, d. In the finite dimensional or Hilbert case, the singular value decomposition can be applied to the matricised tensor.

In Sect. 5.3, the tensorisation is introduced, which maps a vector space (usually without any tensor structure) into an isomorphic tensor space. The artificially constructed tensor structure allows interesting applications. While Sect. 5.3 gives only an introduction into this subject, details about tensorisation will follow in Chap. 14.