tensor product (vector spaces)

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 AβŠ—B one was obliged to chose bases


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


One then represented AβŠ—B relative to this particular choice of bases as the vector space of mappings c:IΓ—J→𝕂. These mappings were called β€œsecond-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 πšβŠ—π›, relative to the chosen bases, as the tensor


In this system, the productsPlanetmathPlanetmath


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


These gave a basis of AβŠ—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 Iβ€² and Jβ€² respectively. Let


be the corresponding change of basis matrices determined by

𝐚iβ€²β€² =βˆ‘i∈I(riβ€²i)⁒𝐚i,iβ€²βˆˆIβ€²
𝐛jβ€²β€² =βˆ‘j∈I(sjβ€²j)⁒𝐛j,jβ€²βˆˆJβ€².

One then stipulated that tensors c:IΓ—J→𝕂 and cβ€²:Iβ€²Γ—J′→𝕂 represent the same element of AβŠ—B if

ci⁒j=βˆ‘iβ€²βˆˆIβ€²jβ€²βˆˆJβ€²(riβ€²i)⁒(sjβ€²j)⁒(cβ€²)i′⁒jβ€² (1)

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


constitute an alternate basis of AβŠ—B, and that the change of basis relations are

𝐚iβ€²β€²βŠ—π›jβ€²β€²=βˆ‘i∈Ij∈J(riβ€²i)⁒(sjβ€²j)⁒𝐚iβŠ—π›j,iβ€²βˆˆIβ€²,jβ€²βˆˆJβ€². (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 by 1,…,n, and also representing 𝐚, albeit relative to a different, but equally unspecified basis. Note that the choice of dummy 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 to ℝ, 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 AβŠ—B and the corresponding representation by tensors relative to a basis β€” compare equations (1) and (2). This relationship is the source of the terminology β€œcontravariant tensor” and β€œcontravariant index”, and I surmise that it is this very medieval pit of darkness and confusion that spawned the present-day notion of β€œcontravariant functorMathworldPlanetmath”.


  1. 1.

    Levi-Civita, β€œThe Absolute Differential Calculus.”

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