3.8.5 Tensor Natural Pairing Tensorial Role

The tensor natural pairing links tensor spaces via bilinear mappings, defining its core role in algebraic structures.

Tensor Natural Pairing Tensorial Role is the recognition that the natural pairing itself is not merely an operation performed on tensors but is, more precisely, the canonical contraction map from the tensor product V* ⊗ V down to the field F, making it a fundamental linear map within tensor algebra rather than an external tool applied to it. Understood this way, the pairing sits inside the algebraic machinery of tensors as the specific (0, 0)-valued contraction that corresponds to the identity endomorphism of V, linking the evaluation operation directly to the structure of linear maps and traces.

The Pairing as a Contraction Map

From Tensor Product to Scalar

The tensor product space V* ⊗ V consists of finite sums of simple tensors f ⊗ v, with f in V* and v in V. There is a unique linear map

c : V^{*} \otimes V \to F

determined by its action on simple tensors, c(f ⊗ v) = f(v), and extended linearly to sums. This map c is called the contraction map, and it is exactly the tensorial packaging of the natural pairing: rather than being a bilinear function of two separate arguments, it is realized as a single linear map on the tensor product space, using the universal property of the tensor product to convert the bilinear pairing into a linear one.

Universal Property Justification

The universal property of the tensor product guarantees that every bilinear map out of V* x V factors uniquely through a linear map out of V* ⊗ V. Applying this universal property to the natural pairing bilinear map produces exactly the contraction map c, showing that the pairing's tensorial role is not an added interpretation but a direct consequence of how tensor products are constructed.

Relation to the Identity Endomorphism

The Identity Tensor

The space V* ⊗ V is canonically isomorphic to the space of linear endomorphisms End(V) = Hom(V, V), via the map sending f ⊗ v to the rank-one operator w ↦ f(w) v. Under this isomorphism, the identity endomorphism id_V corresponds to a specific element of V* ⊗ V, called the identity tensor, expressed in a basis as

{id}_{V} \leftrightarrow \sum_{i = 1}^{n} e^{i} \otimes e_{i}

Trace as Contraction of the Identity Tensor

Applying the contraction map c to the identity tensor recovers the trace of the identity map, which equals the dimension of V:

c (\sum_{i = 1}^{n} e^{i} \otimes e_{i}) = \sum_{i = 1}^{n} e^{i} (e_{i}) = \sum_{i = 1}^{n} 1 = n

demonstrating that the general trace of an endomorphism is itself computed via repeated applications of the same natural pairing that defines the simplest covector-vector evaluation.

The Pairing as the Building Block of General Contractions

Contracting Higher-Rank Tensors

For a general (p, q) tensor T in T^p_q(V) = V^{⊗p} ⊗ (V^*)^{⊗q}, contracting one chosen upper slot against one chosen lower slot applies the contraction map c to that pair of factors while leaving the identity map on all remaining factors. This shows that every tensor contraction, no matter how many indices the surrounding tensor carries, decomposes into the identity map tensored with the single elementary contraction c acting on exactly one V* factor and one V factor.

Iterated Contractions

Fully contracting a (p, p) tensor down to a scalar, such as computing a generalized trace, applies c repeatedly, once for each matched pair of upper and lower indices, and the tensorial role of the natural pairing is what guarantees these repeated applications combine associatively and produce a result independent of the order in which the individual contractions are carried out.

Why the Tensorial Formulation Matters

Linear Maps Are Easier to Compose Than Bilinear Maps

Packaging the pairing as a linear map c : V* ⊗ V -> F, rather than leaving it as a bilinear map on V* x V, allows it to be composed with other linear maps using ordinary function composition, which is essential when building longer chains of tensor operations, such as composing several contractions or combining contraction with other linear transformations of a tensor.

Connecting Abstract Algebra to Coordinate Calculations

The tensorial role of the pairing bridges the abstract, basis-free formulation of tensor algebra with the concrete, index-based formulas used in explicit calculations, since expanding c in a basis and its dual basis immediately recovers the familiar summation formula f_i v^i.

Diagrammatic Summary

The diagram depicts the contraction map c as the linear map realizing the natural pairing on the tensor product space, sending each simple tensor f ⊗ v to the scalar f(v).