12.9.5 Tensor Evaluation Multilinear Consistency

Tensor Evaluation Multilinear Consistency is the property guaranteeing that the scalar or tensor produced by evaluating a tensor on its arguments behaves linearly with respect to each individual argument, and that this linear behavior remains consistent no matter which slot is varied or which basis is used to carry out the computation.

Statement of Multilinear Consistency

Linearity Holding Independently in Every Slot

For a tensor $$A$$ evaluated on a collection of vectors and covectors, multilinear consistency requires that varying any single argument linearly, while holding every other argument fixed, produces an output that varies correspondingly and linearly:

for any scalars $$k$$ and $$l$$ and any vectors or covectors $$u$$ and $$w$$ appropriate to that slot, holding for every one of the tensor's slots individually.

Consistency Across Different Slots

Multilinear consistency additionally requires that this linearity behaves uniformly regardless of which particular slot is being varied, so that the tensor's response to a linear combination in one slot follows exactly the same pattern as its response to a linear combination in any other slot, differing only in which components of the tensor are involved in the relevant contraction.

Basis Independence of the Consistency

Linearity Holds Regardless of Coordinate Representation

Because the componentwise formula for evaluation is built from ordinary multiplication and summation of numbers, which are inherently linear operations, the multilinear behavior of evaluation holds true no matter which basis is used to express the tensor, the vectors, and the covectors involved.

Transformation of Both Sides Together

If a change of basis is applied to the tensor and to all supplied vectors and covectors simultaneously, both sides of the multilinearity identity transform consistently, since the transformation law is itself linear, preserving the equality between the two sides after the change of basis.

Why This Consistency Matters

Justifying the Tensor Result Case

Multilinear consistency underlies why partially evaluating a tensor produces another genuinely multilinear object, since the remaining open slots continue to satisfy the same linearity property with respect to whatever future arguments they eventually receive.

Supporting Reliable Algebraic Manipulation

Because multilinear consistency guarantees that linear combinations of inputs translate predictably into linear combinations of outputs, it underlies safe algebraic manipulations such as expanding an evaluation over a sum of vectors, factoring out a common scalar from one slot, or decomposing an input vector into basis components before evaluation.

Connecting to the Componentwise Formula

The general componentwise formula for evaluation, involving a sum over products of tensor components with vector and covector components, is itself a direct manifestation of multilinear consistency, since each term in that sum depends linearly on exactly one component from each supplied argument.

Illustration