4.11.4 Tensor Multilinear Component Evaluation

Tensor Multilinear Component Evaluation assesses tensor components through multilinear operations, decomposing complex data into structured, interpretable parts.

Tensor Multilinear Component Evaluation is the reduction of the tensor multilinear evaluation operation to a finite arithmetic computation carried out entirely on numerical components, once a basis has been chosen for the underlying vector space and both the tensor and its arguments have been expressed as coordinate arrays. It replaces the abstract pairing of a tensor with vectors and covectors by a concrete sum of products of numbers, giving the evaluation operation a form that can be carried out directly by hand or by a computer.

Setting Up the Component Computation

Choosing a Basis and Its Dual

Let V be a finite-dimensional vector space with basis e_1, ..., e_n, and let e^1, ..., e^n denote the dual basis of V*, satisfying e^i(e_j) = δ^i_j. A type (p, q) tensor T is described, relative to this basis, by its component array T^{i_1 ... i_p}_{j_1 ... j_q}, obtained by evaluating T on all possible combinations of basis vectors and basis covectors.

Expressing the Arguments in Coordinates

Each covector argument α^k and each vector argument v_l supplied to T is expanded in the chosen basis:

α^{k} = \sum_{i = 1}^{n} α_{k}^{i} e^{i}

v_{l} = \sum_{j = 1}^{n} v_{l}^{j} e_{j}

where α_{k,i} denotes the i-th coordinate of α^k and v_l^j denotes the j-th coordinate of v_l, each ranging from 1 to n.

The Component Evaluation Formula

Assembling the Full Sum

Substituting the coordinate expansions of every argument into the multilinear map and using multilinearity to distribute T across each sum produces the component evaluation formula:

T (α^{1}, \dots, v_{q}) = \sum_{i_{1}, \dots, j_{q}}^{⋮} T_{j_{1} \dots j_{q}}^{i_{1} \dots i_{p}} α_{1}^{i_{1}} \dots α_{p}^{i_{p}} v_{1}^{j_{1}} \dots v_{q}^{j_{q}}

where the sum runs over all n^{p+q} combinations of the indices i_1, ..., i_p, j_1, ..., j_q, each ranging independently from 1 to n, and the Einstein summation convention allows the summation symbol itself to be omitted when repeated upper and lower indices are understood to be summed.

Reading the Formula as an Arithmetic Procedure

Practically, the component evaluation formula instructs a direct computation: for every combination of indices, multiply the corresponding component of T by the matching coordinate of each argument, and add up all such products; the result is a single number equal to the tensor evaluated on the given arguments, obtained without any further reference to the abstract vector space V.

Component Evaluation in Low Rank Cases

Rank Zero: Scalars

A type (0, 0) tensor has a single component, with no indices at all, and the component evaluation formula degenerates to the trivial statement that the value of the tensor is simply that one component, unaffected by any arguments since there are none to supply.

Rank One: Vectors and Covectors

For a type (1, 0) tensor, identified with a vector w with components w^i, evaluating it on a single covector α with components α_i reduces to the ordinary dot-product-like sum ∑_i w^i α_i, matching the standard pairing between a vector and a covector.

Rank Two: Matrices

For a type (0, 2) tensor with components T_{ij}, evaluating it on two vectors v and w with components v^i and w^j reduces to the double sum ∑_{i,j} T_{ij} v^i w^j, which is exactly the quadratic form computation familiar from expressing a bilinear form using a matrix and two coordinate vectors.

Complexity and Practical Considerations

Number of Terms in the Sum

The component evaluation formula involves n^{p+q} individual products before summation, so the computational cost of evaluating a tensor grows exponentially with its total rank p + q and polynomially with the dimension n of the underlying space, which is the reason higher-rank tensors are handled with specialized contraction algorithms in practice rather than by expanding the full sum directly.

Consistency with Basis-Free Evaluation

Regardless of which basis is used to carry out the component evaluation, the resulting scalar always agrees with the value obtained from the basis-independent tensor multilinear evaluation operation, since the component formula is derived directly from that operation by substituting coordinate expansions, and does not introduce any dependence on the particular basis chosen beyond the intermediate bookkeeping of components.

Diagrammatic Summary

The diagram shows the component evaluation as repeated multiplication of tensor components by argument coordinates across every combination of indices, summed together to yield the single scalar result.