4.16 Tensor Multilinear Extension Construction

Tensor Multilinear Extension Construction is the general method for building a well-defined multilinear map from data specified only on a spanning or generating set of each argument space, together with the verification steps required to confirm that the resulting map is both well-defined and genuinely multilinear. It generalizes tensor multilinear extension from basis beyond the special case of a basis, addressing the more delicate situation in which the specifying data is given on a set of vectors that may satisfy linear relations among themselves, so that consistency, not merely construction, becomes the central concern.

The General Construction Problem

Specifying Data on a Generating Set

Let V be a finite-dimensional vector space and let g_1, ..., g_m be a spanning set for V, not necessarily linearly independent. Suppose a candidate multilinear map is to be built whose values are prescribed on every combination of the g_r in place of arguments, one prescribed scalar for each combination:

for every combination of indices r_1, ..., r_{p+q} each ranging from 1 to m. The construction problem asks whether, and how, this data extends to a genuine multilinear map defined on all of V and its dual.

The Extension Formula

If the extension exists, it must be given, for arbitrary arguments expanded as v_l = ∑_r c_l^r g_r, by the same kind of weighted sum used in ordinary basis expansion, replacing the basis with the spanning set g_1, ..., g_m and using the prescribed values c(r_1, ..., r_{p+q}) in place of basis values.

The Consistency Requirement

Linear Relations Among the Generators

Because g_1, ..., g_m need not be linearly independent, some vector v may admit more than one expansion in terms of the g_r, differing by a linear combination of the g_r that sums to zero; a genuine multilinear extension exists only if the prescribed values c(r_1, ..., r_{p+q}) are consistent with every such relation, producing the same output regardless of which expansion of v is used.

The Compatibility Condition

Precisely, if ∑_r λ_r g_r = 0 is any linear relation among the generators, then the prescribed data must satisfy

in every slot where the relation is substituted, since this is exactly what additivity and homogeneity would force if a genuine multilinear extension existed; when the generators form a basis, no nontrivial relations exist, so this compatibility condition is automatically satisfied and imposes no restriction, which is why tensor multilinear extension from basis requires no separate consistency check.

Constructing the Extension When Consistency Holds

Well-Definedness of the Resulting Map

If the compatibility condition holds for every linear relation among the generators, the extension formula produces the same output no matter which expansion of an argument in terms of the g_r is used, since any two expansions of the same vector differ by a relation, and the compatibility condition guarantees this difference contributes nothing to the output; the resulting map is therefore well-defined.

Verifying Multilinearity of the Well-Defined Extension

Once well-definedness is established, multilinearity of the extension follows by the same argument used for extension from a basis: the extension formula is linear, to the first power, in the coordinates of each argument relative to the chosen expansions, so additivity and homogeneity hold automatically in every slot.

Reduction to the Basis Case

Choosing a Basis Within the Spanning Set

Any spanning set contains a basis as a subset; restricting the prescribed data to only the combinations involving this basis subset recovers an ordinary basis value assignment, and the extension constructed from the full spanning set data must agree with the extension constructed from this restricted basis data, provided the original spanning set data was consistent.

When the Spanning Set Is Already a Basis

If g_1, ..., g_m happens to be linearly independent, so that m = n and no nontrivial relations exist, the general extension construction reduces exactly to tensor multilinear extension from basis, with no compatibility condition to verify, confirming that the basis case is the simplest and most direct instance of the general construction.

Practical Significance

Building Tensors from Overdetermined Data

The general extension construction is relevant whenever a tensor's data is naturally presented in terms of more vectors than a minimal basis would require, such as a set of generators arising from a physical or geometric description; verifying compatibility ensures that such potentially redundant data still defines a single, unambiguous tensor.

Detecting Inconsistent Data

If the compatibility condition fails for some relation among the generators, no multilinear extension exists at all; this failure serves as a diagnostic indicating that the originally prescribed data was not, in fact, obtainable from any genuine tensor, since a true tensor's values on any spanning set would automatically satisfy every compatibility condition implied by its multilinearity.

Diagrammatic Summary

The diagram shows the general construction proceeding from prescribed data on a spanning set, through a consistency check against every linear relation among the generators, to a well-defined multilinear tensor once that check succeeds.