2012 | OriginalPaper | Chapter
General Techniques
Author : Wolfgang Hackbusch
Published in: Tensor Spaces and Numerical Tensor Calculus
Publisher: Springer Berlin Heidelberg
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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.