4.15.3 Tensor Basis Expansion Evaluation

Tensor Basis Expansion Evaluation is the practical technique of evaluating a tensor on arbitrary arguments by first expanding each argument in a fixed basis and then combining the resulting coordinates with the tensor's already-known basis values, rather than working with the tensor's action on general arguments directly. It is the operational payoff of tensor multilinear basis determination: because the basis values alone determine the tensor completely, any evaluation can be reduced to a lookup in the basis value table followed by a weighted sum.

The Two-Stage Evaluation Procedure

Stage One: Expanding the Arguments

Given a type (p, q) tensor T with known basis values T(e^{i_1}, ..., e_{j_q}) for every combination of basis indices, and given arbitrary arguments α^1, ..., v_q to evaluate T on, the first stage expands every argument in the chosen basis:

producing a coordinate array for each of the p + q arguments, exactly as in the tensor multilinear component evaluation formula.

Stage Two: Combining Coordinates with Basis Values

The second stage forms the weighted sum over all combinations of basis indices, multiplying the known basis value for each combination by the corresponding coordinates drawn from every argument:

with the basis values T(e^{i_1}, ..., e_{j_q}) looked up directly rather than recomputed, since they are already known in advance from the basis value assignment defining the tensor.

Why This Reduction Is Valid

Justification via Additivity and Homogeneity

The correctness of basis expansion evaluation follows directly from the tensor multilinear extension from basis: because the tensor's behavior on arbitrary arguments is entirely forced, through additivity and homogeneity, by its behavior on basis elements, expanding the arguments and recombining them with the basis values reproduces exactly the same result as evaluating the tensor directly on the original, unexpanded arguments.

No Information Lost in the Expansion

Since every vector and covector has a unique coordinate expansion relative to a fixed basis, and since the basis values determine the tensor uniquely by tensor multilinear basis determination, the two-stage procedure loses no information at any step: expanding the arguments and later recombining with the basis value table is a lossless reformulation of direct evaluation.

Practical Advantages of the Technique

Reusing a Fixed Basis Value Table

Once the basis value table for a tensor has been computed or specified, it can be reused for every subsequent evaluation on new arguments, without ever needing to reconstruct or reason about the tensor as an abstract multilinear map again; only the coordinate expansion of the new arguments needs to be computed each time.

Suitability for Computer Representation

Because a tensor's basis value table is a finite array of numbers, and because basis expansion evaluation reduces every evaluation to array lookups combined with sums of products, this technique is the standard method by which tensors are represented and evaluated in numerical software, where tensors are stored directly as their basis value arrays.

Relation to Partial Basis Expansion

Expanding Only Some Arguments

The technique can be applied selectively, expanding only some of the arguments in the basis while leaving others as general vectors or covectors; the resulting expression becomes a sum over fewer indices, weighted by tensor remaining slot maps obtained by fixing the unexpanded arguments through tensor multilinear partial evaluation applied to the basis elements corresponding to the expanded slots.

Building Toward the Full Evaluation Incrementally

Partial basis expansion evaluation, applied to more and more arguments in succession, converges on the full basis expansion evaluation described above once every argument has been expanded; this incremental view connects the technique directly to the slot-by-slot extension procedure used to justify basis determination in the first place.

Diagrammatic Summary

The diagram shows the two ingredients, coordinate-expanded arguments and a pre-computed basis value table, combining into the final scalar evaluation result of the tensor.