4.13 Tensor Multilinear Additivity Property

Tensor Multilinear Additivity Property is the requirement that a multilinear map, when any single argument is replaced by a sum of two vectors or covectors, produces an output equal to the sum of the outputs obtained by substituting each summand separately, holding every other argument fixed. It is one of the two component conditions, alongside homogeneity, that together constitute linearity in each slot, and it is precisely the condition responsible for the word "linear" applying separately to every argument of a multilinear map.

Formal Statement of Additivity

Additivity in a Single Slot

For a type (p, q) tensor T on a vector space V, additivity in the l-th covariant slot states that for any vectors u, w ∈ V and any fixed choices of the remaining arguments,

where the argument in the l-th slot is replaced by u + w on the left, and the same slot is filled separately by u and by w on the right, with every other argument, marked by the placeholders ⋯, held identical across all three evaluations. An analogous statement holds for each of the p contravariant slots, with covectors in place of vectors.

Additivity Applies Slot by Slot, Not Jointly

Additivity is required to hold in each slot individually, with all other slots fixed; it is not, in general, required that T(u_1 + w_1, ..., u_q + w_q) equal any simple combination of the 2^q possible evaluations obtained by choosing u_l or w_l independently in each slot, although such an expanded identity does follow as a consequence of applying single-slot additivity repeatedly across all slots at once.

Additivity as Half of Linearity

The Companion Property of Homogeneity

Additivity alone does not yet constitute full linearity in a slot; it must be paired with homogeneity, the requirement that scaling an argument by a constant λ scales the output by the same constant:

Together, additivity and homogeneity in a slot constitute exactly the statement that T, viewed as a function of that one slot alone with the others fixed, is a linear map, and it is the conjunction of both properties across every slot that defines multilinearity.

Additivity Implies Homogeneity by Rational Scalars

Over the rational numbers, additivity alone already forces homogeneity for rational scalars, since any rational multiple can be built from repeated addition and the definition of additive inverses; homogeneity for arbitrary real or complex scalars, however, is not derivable from additivity alone in complete generality and is typically imposed as a separate requirement, particularly when continuity or additional structure is not assumed.

Consequences of Additivity

Expansion Over a Sum of Basis Vectors

Additivity, applied repeatedly, is what permits any argument to be expanded in a basis and distributed term by term across the tensor, which is the mechanism underlying the tensor multilinear component evaluation formula: writing an argument as a sum of scaled basis vectors and applying additivity slot by slot decomposes the evaluation into a sum of simpler evaluations on individual basis vectors.

Additivity and the Tensor Product Universal Property

The universal property characterizing the tensor product of vector spaces is stated directly in terms of multilinear maps satisfying additivity, alongside homogeneity, in each argument; any multilinear map failing additivity in some slot would not correspond to a well-defined linear map out of the tensor product, since the tensor product is constructed precisely to be the universal recipient of maps that are additive and homogeneous in each factor separately.

Additivity Under Partial Evaluation

Preservation of Additivity in the Remaining Slots

When some slots of T are fixed through partial evaluation, the resulting tensor remaining slot map continues to satisfy additivity in each of its still-open slots, since additivity in a slot depends only on the behavior of T in that slot with the rest held fixed, and fixing other slots does not interfere with this behavior in any way.

Additivity in the Fixed Arguments Themselves

Separately, the reduced arity result produced by partial evaluation depends additively on each fixed argument as well, which is exactly the content of tensor partial evaluation linearity; additivity in the fixed slots and additivity in the remaining slots are two aspects of the same underlying multilinear structure of T.

Verifying Additivity in Practice

Checking Additivity Componentwise

In component form, additivity in a slot is automatically satisfied whenever the evaluation formula is a sum of products in which the coordinates of that slot's argument appear linearly, to the first power, and are not combined with each other by any operation other than addition and scalar multiplication; any component formula respecting this pattern is guaranteed to be additive in that slot.

Failure of Additivity as a Diagnostic

If a proposed map fails additivity in some slot, for instance because it involves a squared coordinate or a product of two coordinates from the same argument, this failure is a direct indication that the map is not multilinear and therefore cannot be represented as a tensor of any type, since tensors are defined precisely as maps satisfying additivity, together with homogeneity, in every slot.

Diagrammatic Summary

The diagram shows the defining behavior of additivity, in which supplying a sum of two arguments to one slot of the tensor produces the same result as evaluating the tensor separately on each summand and adding the two outputs together.