tensor product (vector spaces) (original) (raw)

Definition. The classical conception of the tensor productPlanetmathPlanetmathPlanetmath operationMathworldPlanetmath involved finite dimensional vector spacesMathworldPlanetmath A, B, say over a field 𝕂. To describe the tensor product A⊗B one was obliged to chose bases

of A and B indexed by finite setsMathworldPlanetmath I and J, respectively, and represent elements of𝐚∈A and 𝐛∈B by their coordinatesPlanetmathPlanetmath relative to these bases, i.e. as mappings a:I→𝕂 and b:J→𝕂such that

𝐚=∑i∈Iai⁢𝐚i,𝐛=∑j∈Jbj⁢𝐛j.

One then represented A⊗B relative to this particular choice of bases as the vector space of mappings c:I×J→𝕂. These mappings were called “second-order contravariant tensors” and their values were customarily denoted by superscripts, a.k.a. contravariant indices:

The canonical bilinear multiplicationPlanetmathPlanetmath (also known as outer multiplication)

was defined by representing 𝐚⊗𝐛, relative to the chosen bases, as the tensor

In this system, theproductsPlanetmathPlanetmath

were represented by basic tensors, specified in terms of the Kronecker deltasMathworldPlanetmath as the mappings

(i′,j′)↦δii′⁢δjj′,i′∈I,j′∈J.

These gave a basis of A⊗B.

The construction is independent of the choice of bases in the following sense. Let

be different bases of A and B with indexing sets I′and J′ respectively. Let

be the corresponding change of basis matrices determined by

𝐚i′′ =∑i∈I(ri′i)⁢𝐚i,i′∈I′
𝐛j′′ =∑j∈I(sj′j)⁢𝐛j,j′∈J′.

One then stipulated that tensors c:I×J→𝕂 andc′:I′×J′→𝕂 represent the same element ofA⊗B if

ci⁢j=∑i′∈I′j′∈J′(ri′i)⁢(sj′j)⁢(c′)i′⁢j′ (1)

for all i∈I,j∈J. This relationMathworldPlanetmathPlanetmath corresponds to the fact that the products

constitute an alternate basis of A⊗B, and that the change of basis relations are

𝐚i′′⊗𝐛j′′=∑i∈Ij∈J(ri′i)⁢(sj′j)⁢𝐚i⊗𝐛j,i′∈I′,j′∈J′. (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 𝐚 of a vector space, one was obliged to contemplate a symbol 𝐚i, which signified a list of real numbers indexed by 1,2,…,n, and which was understood to represent 𝐚 relative to some specified, but unnamed basis.

What makes this notation truly lethal is the fact a symbol 𝐚j was taken to signify an alternate list of real numbers, also indexed by1,…,n, and also representing 𝐚, albeit relative to a different, but equally unspecified basis. Note that the choice ofdummy variables make all the differencePlanetmathPlanetmath. Any sane system of notation would regard the expression

as representing a list of n 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 𝐚i as a mapping from some finite index setMathworldPlanetmath I toℝ, whereas 𝐚j is interpreted as a mapping from another finite index set J (of equal cardinality) to ℝ.

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 A⊗B and the corresponding representation by tensors relative to a basis — compare equations (1) and (2). This relationship is the source of the terminology “contravariant tensor” and “contravariant index”, and I surmise that it is this very medieval pit of darkness and confusion that spawned the present-day notion of “contravariant functorMathworldPlanetmath”.

References.

    1. Levi-Civita, “The Absolute Differential Calculus.”