tensor product (vector spaces)
Definition. The classical conception of the tensor product
operation
involved finite dimensional vector spaces
, , say over
a field . To describe the tensor product one
was obliged to chose bases
of and indexed
by finite sets and , respectively, and represent elements of
and by their coordinates
relative to these bases,
i.e. as mappings and
such that
One then represented relative to this particular choice of bases as the vector space of mappings . These mappings were called βsecond-order contravariant tensorsβ and their values were customarily denoted by superscripts, a.k.a. contravariant indices:
The canonical bilinear
multiplication (also known as outer multiplication)
was defined by representing , relative to the chosen bases, as the tensor
In this system, the
products
were represented by basic
tensors, specified in terms of the Kronecker deltas as the mappings
These gave a basis of .
The construction is independent of the choice of bases in the following sense. Let
be different bases of and with indexing sets and respectively. Let
be the corresponding change of basis matrices determined by
One then stipulated that tensors and represent the same element of if
(1) |
for all . This relation corresponds to the fact that
the products
constitute an alternate basis of , and that the change of basis relations are
(2) |
Notes. Historically, the tensor product was called the outer product, and has its origins in the absolute differential calculus (the theory of manifolds). The old-time tensor calculus is difficult to understand because it is afflicted with a particularly lethal notation that makes coherent comprehension all but impossible. Instead of talking about an element of a vector space, one was obliged to contemplate a symbol , which signified a list of real numbers indexed by , and which was understood to represent relative to some specified, but unnamed basis.
What makes this notation truly lethal is the fact a symbol was
taken to signify an alternate list of real numbers, also indexed by
, and also representing , albeit relative to a
different, but equally unspecified basis. Note that the choice of
dummy variables make all the difference. Any sane system of
notation would regard the expression
as representing a list of symbols
However, in the classical system, one was strictly forbidden from using
because where, after all, is the all important dummy variable to indicate choice of basis?
Thankfully, it is possible to shed some light onto this confusion (I
have read that this is credited to Roger Penrose) by interpreting the
symbol as a mapping from some finite index set to
, whereas is interpreted as a mapping from another
finite index set (of equal cardinality) to .
My own surmise is that the source of this notational difficulty stems from the reluctance of the ancients to deal with geometric objects directly. The prevalent superstition of the age held that in order to have meaning, a geometric entity had to be measured relative to some basis. Of course, it was understood that geometrically no one basis could be preferred to any other, and this leads directly to the definition of geometric entities as lists of measurements modulo the equivalence engendered by changing the basis.
It is also worth remarking on the contravariant nature of the
relationship between the actual elements of and the
corresponding representation by tensors relative to a basis β compare
equations (1) and (2).
This relationship is the source of the
terminology βcontravariant tensorβ and βcontravariant indexβ, and
I surmise that it
is this very medieval pit of darkness and confusion that spawned the
present-day notion of βcontravariant functorβ.
References.
-
1.
Levi-Civita, βThe Absolute Differential Calculus.β
Title | tensor product (vector spaces) |
---|---|
Canonical name | TensorProductvectorSpaces |
Date of creation | 2013-03-22 12:21:40 |
Last modified on | 2013-03-22 12:21:40 |
Owner | rmilson (146) |
Last modified by | rmilson (146) |
Numerical id | 10 |
Author | rmilson (146) |
Entry type | Definition |
Classification | msc 15A69 |
Synonym | tensor product |
Related topic | OuterMultiplication |