4.11 Tensor Multilinear Evaluation Operation

Tensor Multilinear Evaluation Operation is the operation by which a tensor, viewed as a multilinear map, is fed a specific choice of vector and covector arguments and returns the resulting scalar value in the base field. It is the fundamental operation that connects the abstract algebraic description of a tensor as an element of a tensor product space with its concrete role as a function, since it is precisely this evaluation that a tensor must perform, linearly in each argument, in order to justify being called a multilinear map in the first place.

The Operation in Its Basic Form

Domain and Codomain of Evaluation

For a type (p, q) tensor T on a finite-dimensional vector space V, the evaluation operation takes p covectors α^1, ..., α^p ∈ V* together with q vectors v_1, ..., v_q ∈ V and produces a single scalar:

where F is the base field. The operation accepts exactly p + q arguments, in the fixed order dictated by the type of T, and its output has no further indices or free slots, since a scalar has been produced.

Multilinearity of the Operation

The defining property of the evaluation operation is that it is linear separately in each of its p + q arguments, holding all others fixed. This means that for any argument slot, scaling the input by a constant scales the output by the same constant, and adding two inputs in that slot adds the two corresponding outputs, while every other argument remains unchanged.

Evaluation via Basis Contraction

Reducing Evaluation to Components

When T is expressed in components relative to a basis e_1, ..., e_n of V and its dual basis e^1, ..., e^n of V*, and each argument is expanded in the corresponding basis, the evaluation operation reduces to a finite sum over all combinations of basis indices, using the Einstein summation convention:

Here each repeated index, appearing once as an upper index of T and once as a lower index of a coordinate, or vice versa, is summed over its full range from 1 to n, and the sum collapses to the single scalar value of the evaluation.

Pairing Basis Vectors and Basis Covectors

The individual products appearing in this sum rest on the natural pairing between a basis covector e^i and a basis vector e_j, which returns the Kronecker delta:

which is 1 when i equals j and 0 otherwise. It is this pairing, extended bilinearly, that underlies the contraction between each upper index of T and a covector argument, and between each lower index of T and a vector argument.

Evaluation as a Sequence of Partial Contractions

Filling One Slot at a Time

The evaluation operation may also be carried out one argument at a time, treating T as a curried function. Supplying only the first argument α^1 produces an intermediate tensor of type (p - 1, q), since one contravariant slot has been consumed while the remaining slots are untouched:

where the placeholders ⋅ mark the slots left open for the remaining arguments. Repeating this process for each subsequent argument yields, after all p + q slots are filled, the same scalar value produced by the full simultaneous evaluation, since multilinearity guarantees the result does not depend on the order in which the slots are filled.

Equivalence with Tensor Contraction

This slot-by-slot evaluation is a special case of tensor contraction, in which each contracted index pairs an upper index of T with the corresponding index of a covector argument, or a lower index of T with the corresponding index of a vector argument, and the contraction operation is applied repeatedly until no free indices remain.

Properties of the Evaluation Operation

Independence from the Choice of Basis

Although the component formula for evaluation refers to a specific basis, the resulting scalar value is independent of that choice, since T as an abstract multilinear map, and the arguments as abstract vectors and covectors, do not depend on any coordinate system; changing the basis changes the individual components of T and of the arguments simultaneously, in a way that leaves their contracted product invariant.

Evaluation as a Bilinear Pairing Between Tensor Spaces

The evaluation operation can itself be viewed as a canonical bilinear pairing between the space of type (p, q) tensors and the space of type (q, p) tensors formed from p copies of V and q copies of V*, since supplying p covectors and q vectors to T is equivalent to pairing T against the tensor product of those covectors and vectors.

Diagrammatic Summary

The diagram shows the tensor T receiving covector arguments in its upper slots and vector arguments in its lower slots, with the evaluation operation collapsing all of them together to produce a single scalar output.