2.20.4 Tensor Vector Covector Pairing

Tensor Vector Covector Pairing explores how tensors interact with vectors and covectors through bilinear mappings, foundational in differential geometry and physics.

Tensor Vector Covector Pairing is the bilinear evaluation map that combines a covector from V* with a vector from V to produce a scalar, treated here as an object of study in its own right, with its own bilinearity properties, nondegeneracy, matrix representation, and role as the seed operation from which every tensor contraction is built. Where the dual vector space context introduces the pairing as background structure and the covector space context treats V* as a whole space, the vector covector pairing isolates the evaluation map itself, ⟨ , ⟩ : V* × V → F, and studies its algebraic properties as a bilinear form defined without reference to any metric.

Definition and Basic Properties

The Pairing as a Bilinear Map

The natural pairing sends a covector f in V* and a vector v in V to the scalar f(v), and is written ⟨f, v⟩. It is linear in each argument separately:

⟨ α f + β g, v ⟩ = α ⟨ f, v ⟩ + β ⟨ g, v ⟩

⟨ f, γ u + δ v ⟩ = γ ⟨ f, u ⟩ + δ ⟨ f, v ⟩

This bilinearity is exactly the property required for the pairing to serve as a (1, 1)-type multilinear map, taking one covector argument and one vector argument.

Nondegeneracy

The pairing is nondegenerate in both arguments: if ⟨f, v⟩ = 0 for every v in V, then f must be the zero covector, and if ⟨f, v⟩ = 0 for every f in V*, then v must be the zero vector. The first statement is essentially the definition of a covector being zero; the second follows because for any nonzero v, it can be extended to a basis, and the corresponding dual basis covector e^i satisfies ⟨e^i, v⟩ = 1 ≠ 0. Nondegeneracy is what makes the pairing a perfect pairing, meaning it induces isomorphisms V* ≅ (V)^* and V ≅ (V*)^* in the expected way, without collapsing any nonzero vector or covector to a functional that acts trivially on everything.

Matrix Representation of the Pairing

Coordinate Form

Once a basis e_1, ..., e_n of V and the corresponding dual basis e^1, ..., e^n of V* are fixed, the pairing takes the coordinate form

⟨ f, v ⟩ = f_{i} v^{i}

using the Einstein summation convention, where f_i are the components of f in the dual basis and v^i are the components of v in the basis. In matrix language, this is the product of a row vector (f_1, ..., f_n) and a column vector (v^1, ..., v^n)^T.

The Pairing Matrix Is the Identity in Dual Bases

If the components of f and v are expressed relative to a basis and its dual, the "pairing matrix," the array of values ⟨e^i, e_j⟩, is simply the identity matrix, a direct restatement of the biorthogonality condition defining the dual basis. This is the special feature that distinguishes a dual-basis pairing from a general bilinear form such as an inner product, whose matrix in an arbitrary basis need not be the identity or even diagonal.

The Pairing as the Prototype for Tensor Contraction

Single Contraction

Applying the pairing to a covariant index and a contravariant index of two separate tensors, or of a single mixed tensor, is precisely the operation known as contraction. For a (1, 0) tensor v and a (0, 1) tensor f, contraction reduces to the scalar ⟨f, v⟩ directly; for higher-rank tensors, contraction applies the same pairing to one chosen pair of indices while leaving the remaining indices untouched.

C (T) = T_{i}^{i}

illustrates contraction on a (1, 1) tensor T, where the repeated index i is summed, producing the trace-like scalar T^i_i, exactly the pairing applied to T regarded as a linear map identified with an element of V ⊗ V*.

Trace as a Special Case

The trace of a linear operator A : V → V, when A is regarded as an element of V ⊗ V* via the identification of (1,1)-tensors with endomorphisms, is precisely the contraction produced by the vector-covector pairing applied to that single pair of indices. This shows the pairing is not merely a two-argument evaluation but the elementary building block from which the trace, and more generally every tensor contraction, is assembled.

Naturality Under Linear Maps

Pairing Compatibility With Pullback and the Map Itself

For a linear map T : V → W, the pairing on W* × W and the pairing on V* × V are related by the defining identity of the pullback map T* : W* → V*:

⟨ T^{*} g, v ⟩ = ⟨ g, T (v) ⟩

for every g in W* and v in V. This identity is exactly what defines the pullback (also called the transpose or adjoint) of a linear map, using the pairing as the compatibility condition rather than any coordinate computation, and it shows the pairing behaves naturally, in the technical sense, with respect to all linear maps between vector spaces.

Contrast With an Inner Product

Unlike an inner product, the vector-covector pairing requires no positive definiteness, no symmetry, and no single underlying space, since its two arguments live in genuinely different spaces, V* and V. This makes the pairing more elementary than an inner product: an inner product on V can be built from the pairing together with a chosen isomorphism V ≅ V*, but the pairing itself exists for any vector space without further structure, which is why it, rather than an inner product, is the operation used to define contraction throughout general tensor algebra.