4.2 Tensor Bilinear Map Structure

Tensor Bilinear Map Structure is the special case of the multilinear map correspondence in which a tensor of rank two is realized as a map taking exactly two arguments, linear in each separately, and it forms the simplest nontrivial instance from which the general theory of tensor multilinear maps is developed, encompassing bilinear forms, inner products, and rank-2 mixed tensors alike.

Definition and Basic Types

The Bilinear Map Condition

A map B: X × Y → ℝ, where X and Y are each either V or V*, is bilinear if it is linear in each argument separately:

The Three Tensor Types from Two Arguments

Depending on whether each slot accepts a vector or a covector, bilinear maps correspond to three distinct rank-2 tensor types: a map V × V → ℝ corresponds to a type (0, 2) tensor, commonly called a bilinear form; a map V* × V* → ℝ corresponds to a type (2, 0) tensor; and a map V* × V → ℝ or V × V* → ℝ corresponds to a type (1, 1) tensor, which is naturally identified with a linear endomorphism of V.

Coordinate Representation

Matrix of a Bilinear Form

Fixing a basis e_1, ..., e_n of V, a type (0, 2) bilinear form B is completely determined by the n × n matrix of its values on basis pairs,

and its output on arbitrary vectors u = u^i e_i and v = v^j e_j is recovered as

in matrix notation, exhibiting the bilinear map structure as an ordinary quadratic-form-style matrix pairing.

Type (1, 1) Tensors as Linear Operators

A type (1, 1) tensor T, seen as a bilinear map V* × V → ℝ, corresponds to the linear operator A: V → V defined by T(ω, v) = ω(A(v)), where the components T^i_j are exactly the matrix entries of A in the chosen basis. This identification is the bilinear-map-level explanation for why every square matrix can be read simultaneously as a linear operator and as a mixed rank-2 tensor.

Symmetric and Antisymmetric Bilinear Structure

Symmetric Bilinear Maps

A bilinear form B on V × V is symmetric if B(u, v) = B(v, u) for all u, v, equivalently if its component matrix satisfies B_ij = B_ji. Symmetric bilinear forms generalize the ordinary dot product and give rise to associated quadratic forms Q(v) = B(v, v).

Antisymmetric Bilinear Maps

A bilinear form is antisymmetric, or alternating, if B(u, v) = -B(v, u), equivalently B_ij = -B_ji, and B(v, v) = 0 for every v. Antisymmetric bilinear forms are the rank-2 case of alternating tensors, and every bilinear form decomposes uniquely into symmetric and antisymmetric parts,

Bilinear Maps and the Tensor Product

Universal Factorization

The bilinear map structure is the concrete two-argument instance of the tensor product's universal property: every bilinear map B: V × V → ℝ factors uniquely as B = β ∘ ⊗, where ⊗: V × V → V ⊗ V is the canonical bilinear map (u, v) ↦ u ⊗ v and β: V ⊗ V → ℝ is a uniquely determined linear map. This factorization is the precise sense in which V ⊗ V is the universal recipient of bilinear maps out of V × V.

Composition and Transformation

If f: V → W is linear and B is a bilinear form on W, the pullback fB, defined by (fB)(u, v) = B(f(u), f(v)), is again bilinear on V, with matrix representation A^T B A where A is the matrix of f. This transformation law is the bilinear-map instance of the general pullback formula for covariant tensors, showing directly how the rank-2 case governs the pattern followed by pullback at every higher rank.

Nondegeneracy and Rank

Rank of a Bilinear Form

The rank of a bilinear form B is the rank of its matrix representation B_ij, equivalently the dimension of V minus the dimension of its radical, the subspace of vectors v such that B(v, w) = 0 for all w. A bilinear form is nondegenerate exactly when its rank equals dim(V), in which case it induces an isomorphism between V and V* via v ↦ B(v, ·), the mechanism underlying the musical isomorphisms used to raise and lower indices.