Tensors: Geometry and Applications J.M. Landsberg - Texas A&M ...

Multilinear algebra
Chapter 2
This chapter approaches multilinear algebra from a geometric perspective.
If X = (a i s) is a matrix, one is not so much interested in the collection
of numbers that make up X, but what X represents and what qualitative
information can be extracted from the entries of X. For this reason and
others, in §2.3 an invariant definition of tensors is given and its utility is
explained, especially in terms of groups acting on spaces of tensors. Before
that, the chapter begins in §2.1 with a collection of exercises to review facts
from linear algebra that will be important in what follows. For those readers
needing a reminder of the basic definitions in linear algebra, they are given
in an appendix §2.9. Basic definitions regarding group actions that will be
needed throughout are stated in §2.2. In §2.4, rank and border rank of tensors
are defined, Strassen’s algorithm is revisited, and basic results about
rank are established. A more geometric perspective on the matrix multiplication
operator in terms of contractions is given in §2.5. Among subspaces
of spaces of tensors, the symmetric and skew-symmetric tensors discussed
in §2.6 are distinguished, not only because they are the first subspaces one
generally encounters, but all other natural subspaces may be built out of
symmetrizations and skew-symmetrizations. As a warm up for the detailled
discussions of polynomials that appear later in the book, polynomials on
the space of matrices are discussed in §2.7. In §2.8, V ⊗3 is decomposed as
a GL(V )-module, which serves as an introduction to Chapter 6.
There are three appendices to this chapter. As mentioned above, in
§2.9 basic definitions are recalled for the reader’s convenience. §2.10 reviews
Jordan and rational canonical forms. Wiring diagrams, a useful pictorial
tool for studying tensors are introduced in §2.11.
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