4.11.1 Tensor Multilinear Input Tuple

A multilinear input tuple in tensor algebra structures data inputs linearly across multiple dimensions, enabling tensor operations through coordinated scalar mappings.

Tensor Multilinear Input Tuple is the ordered collection of arguments supplied to a multilinear map in a single evaluation, consisting of exactly as many covectors and vectors as the type of the tensor demands, arranged in the fixed order the tensor's slots expect. It is the object that gets fed into the tensor multilinear evaluation operation, and its structure, both the number of entries and the order in which they appear, is what determines which slot of the tensor each argument is paired against.

Structure of the Input Tuple

Composition for a Type (p, q) Tensor

For a type (p, q) tensor T on a vector space V, the input tuple consists of p covectors drawn from V* followed by q vectors drawn from V:

(α^{1}, \dots, α^{p}, v_{1}, \dots, v_{q}) \in \times_{i = 1}^{p} V^{*} \times \times_{j = 1}^{q} V

This Cartesian product is precisely the domain of T when T is regarded as a multilinear map, and any element of it constitutes a valid input tuple for T.

Fixed Order and Slot Assignment

The order of entries within the tuple is not arbitrary: the tensor's type fixes which position corresponds to which kind of index, so the first p entries are always paired against the p contravariant slots of T, and the last q entries are always paired against the q covariant slots. Two input tuples that contain the same arguments but in a different order are, in general, distinct inputs, since multilinear maps need not be symmetric under a permutation of arguments of the same kind unless the tensor is specifically constructed to be so.

The Input Tuple as an Element of a Product Space

Cartesian Product Versus Tensor Product

The input tuple lives in the Cartesian product of the argument spaces, not their tensor product; this distinction matters because the Cartesian product is the natural domain of a multilinear function of several variables, while the tensor product is the space in which the tensor T itself, viewed as a vector, resides. The evaluation operation is exactly the mechanism that bridges these two different product constructions, converting an input tuple in the Cartesian product into a scalar by pairing it against T.

Component Representation of the Tuple

Once bases are fixed for V and V*, each entry of the input tuple decomposes into coordinates, and the full tuple can be described by a rectangular array of scalar coordinates, one row for each entry and one column for each basis index, which together with the components of T determine the scalar output of the evaluation operation through the summation formula for tensor contraction.

Varying One Entry of the Tuple

Partial Tuples and Multilinearity

Fixing every entry of the input tuple except one produces a partial tuple with a single open slot, and the multilinearity of T guarantees that the map from that one remaining argument to the resulting scalar is linear:

T (α^{1}, \dots, λ α^{k} + β, \dots, v_{q}) = λ T (\dots, α^{k}, \dots) + T (\dots, β, \dots)

This slot-wise linearity is the property that gives the "multi" in multilinear its precise meaning: the tuple as a whole need not depend linearly on all its entries simultaneously, since scaling every entry at once scales the output by the product of the scaling factors, but each individual slot behaves linearly when the rest of the tuple is held fixed.

Building the Tuple Incrementally

Because of this slot-wise behavior, an input tuple can be assembled and evaluated incrementally, one entry at a time, with each partial evaluation producing a lower-rank tensor that still awaits the remaining entries, until the full tuple has been supplied and a scalar remains.

Special Kinds of Input Tuples

Basis Input Tuples

An input tuple whose entries are all drawn from a fixed basis of V or its dual basis of V*, one entry per slot, is called a basis input tuple, and the scalars produced by evaluating T on every possible basis input tuple are exactly the components of T relative to that basis.

Diagonal Input Tuples

An input tuple in which the same vector or covector is repeated across several slots of the same kind is called a diagonal input tuple, and evaluating T on such tuples is the operation used to extract quadratic and higher-degree forms from a tensor, such as recovering the norm-squared of a vector from a type (0, 2) tensor by supplying the same vector twice.

Diagrammatic Summary

The diagram shows the input tuple arranged with covector entries first and vector entries second, matching the fixed order expected by the slots of the tensor T, which the evaluation operation then contracts against the tuple.