4.22.1 Tensor Multilinear Component Array

A tensor multilinear component array generalizes vectors and matrices, organizing multi-dimensional data to represent complex linear relationships across indices.

Tensor Multilinear Component Array is the array of scalars representing a general multilinear map f: V₁ × ... × Vₙ → W, including the case of vector-valued output, obtained by evaluating f on every tuple of basis vectors from the input spaces and recording the resulting output vector's own coordinates against a chosen basis of W. Unlike the component array of a scalar-valued multilinear form, this array carries an additional index tracking the output coordinate, reflecting that the result of f is itself a vector rather than a single number.

Construction Including the Output Index

One Extra Index Beyond the Input Indices

Given bases {e^{(1)}_{i₁}}, ..., {e^{(n)}_{iₙ}} of the input spaces and a basis {u_k} of W, the component array of f is

T_{i_{1} \dots i_{n}}^{k} = the coefficient of u_{k} in f (e_{i_{1}}^{(1)}, \dots, e_{i_{n}}^{(n)})

so that f(e^{(1)}_{i₁},...,e^{(n)}_{iₙ}) = ∑ₖ T^k_{i₁...iₙ} uₖ. The array T has n "input" indices, one for each argument slot, and one "output" index k ranging over the basis of W, distinguishing it from the purely input-indexed array associated with a scalar-valued multilinear form.

Total Size of the Array

If each input space Vᵢ has dimension dᵢ and W has dimension p, the array T has d₁ × d₂ × ... × dₙ × p entries in total, one more factor than the scalar-valued case, accounting for the dimension of the codomain.

Reconstruction Formula

Recovering the Vector-Valued Output

Writing each input vector vⱼ = ∑ᵢ vⱼⁱ e^{(j)}_i in coordinates, multilinearity gives

f (v_{1}, \dots, v_{n}) = \sum k (\sum i_{1} \dots \sum i_{n} v_{1}^{i_{1}} \dots v_{n}^{i_{n}} T_{i_{1} \dots i_{n}}^{k}) u_{k}

with the outer sum over k assembling the final output vector coordinate by coordinate, and the inner sum, for each fixed k, matching exactly the reconstruction formula already familiar from scalar-valued multilinear forms, applied separately to each output coordinate.

Reduction to the Scalar-Valued Case

For fixed k, the slice T^k_{i₁...iₙ} (varying only the input indices, output index held fixed) is precisely the component array of the scalar-valued multilinear form obtained by composing f with the k-th coordinate functional on W; the general multilinear component array is therefore a bundle of p ordinary scalar-valued component arrays, one for each output coordinate, matching the "stack of matrices" picture already familiar from the vector-valued matrix case at arity two.

Examples

The Cross Product

The cross product ×: R³ × R³ → R³ has component array T^k_{ij} = ε_{ijk}, the Levi-Civita symbol, with two input indices and one output index, all three ranging over {1,2,3}; this is among the most familiar instances of a vector-valued multilinear component array outside the purely scalar-valued case.

Matrix Multiplication as a Bilinear Map

Viewing matrix multiplication as a bilinear map Hom(V,U) × Hom(U,W) → Hom(V,W), its component array, once bases are fixed for all the relevant Hom-spaces, is a four-index object built from Kronecker deltas, reducing in the standard matrix-index convention to the familiar rule that the (i,k) entry of a product matrix sums over the shared index j between the two factor matrices.

General Linear Combinations of Bilinear Operations

Any vector-valued bilinear operation expressible as several independent scalar bilinear forms combined into a vector, such as constructing a new vector from several inner products against fixed vectors, has a component array that decomposes exactly into that many independent scalar-valued component arrays, one per output coordinate.

Basis Dependence of the Full Array

Transformation Under Change of Basis on Any Factor

Changing the basis of any input space Vⱼ transforms the array by contracting the corresponding input index against the change-of-basis matrix for that factor, exactly as in the scalar-valued case; changing the basis of the output space W additionally transforms the output index k by contraction against the change-of-basis matrix for W, a transformation not present at all for scalar-valued forms, since there W = F has no basis to change beyond scalar multiplication by a unit.

Combined Transformation Law

Under simultaneous changes of basis in every input space and in W, the array transforms by one contraction per index, n contractions for the input indices and one additional contraction for the output index, giving the general transformation law governing a type of tensor with n covariant indices (from the inputs) and one contravariant index (from the vector-valued output), situating the general multilinear component array within the classical type (p,q) classification of tensors.

Practical Role

Storing General Multilinear Maps for Computation

The full component array, including the output index, is the representation used whenever a vector-valued multilinear operation, such as a cross product, a bilinear pairing feeding into a vector space of results, or a multilinear differential operator with vector output, must be implemented or evaluated numerically, since it provides the complete data needed to compute the output vector's coordinates directly from the input vectors' coordinates via the reconstruction formula.

Verifying Multilinearity From the Array

A candidate array T^k_{i₁...iₙ} automatically defines a multilinear map via the reconstruction formula regardless of its specific entries, since the formula is linear in each set of input coordinates separately by construction; no additional multilinearity check on the array itself is needed beyond confirming it has the correct index structure and dimensions.