4.14 Tensor Multilinear Homogeneity Property

Tensor Multilinear Homogeneity Property is the requirement that a multilinear map, when any single argument is scaled by a constant from the base field, produces an output scaled by exactly that same constant, holding every other argument fixed. It is the second of the two component conditions, alongside additivity, that together constitute linearity in each slot, and it is the property responsible for the precise, uniform way a tensor's output tracks any rescaling of a single input.

Formal Statement of Homogeneity

Homogeneity in a Single Slot

For a type (p, q) tensor T on a vector space V over a field F, homogeneity in the l-th covariant slot states that for any vector v ∈ V, any scalar λ ∈ F, and any fixed choice of the remaining arguments,

where the argument in the l-th slot is scaled by λ on the left, and the entire output is scaled by the same λ on the right, with every other argument, marked by the placeholders ⋯, held identical across both sides. An analogous statement holds for each of the p contravariant slots, with a covector scaled by λ in place of a vector.

Homogeneity Applies Slot by Slot

Like additivity, homogeneity is required to hold in each slot individually, with all other slots fixed; scaling several arguments at once by different scalars produces an output scaled by the product of all those scalars, a consequence obtained by applying single-slot homogeneity once for each scaled argument in turn, not a separate, independently imposed condition.

Homogeneity as Half of Linearity

The Companion Property of Additivity

Homogeneity alone does not yet constitute full linearity in a slot; it must be paired with the tensor multilinear additivity property, the requirement that summing two arguments in a slot sums the corresponding outputs. 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 the conjunction of both properties across every slot defines multilinearity.

Special Cases of the Scalar Multiplier

Setting λ = 1 in the homogeneity identity gives the trivial statement T(..., v, ...) = T(..., v, ...), while setting λ = 0 gives T(..., 0, ...) = 0, showing that homogeneity forces the output to vanish whenever any single argument is the zero vector or zero covector, regardless of what the other arguments happen to be.

Consequences of Homogeneity

Vanishing on a Zero Argument

The fact that T(..., 0, ...) = 0 whenever any one slot receives the zero element is a direct and immediate consequence of homogeneity with λ = 0; this is a distinguishing feature of multilinear maps not shared by general functions of several variables, which need not vanish merely because one input is zero.

Scaling Behavior Under Repeated Application

Applying homogeneity to more than one slot at once shows that scaling the l-th argument by λ and the m-th argument by μ, with l ≠ m, scales the total output by λμ, since each scaling is applied independently and their effects multiply, reflecting the multilinear, rather than merely linear, structure of T across several slots simultaneously.

Homogeneity and the Component Description

Direct Proportionality of Component Terms

In component form, relative to a basis, homogeneity in slot l corresponds to each term of the component evaluation formula containing the coordinate v_l^{j_l}} to exactly the first power; scaling this coordinate by λ scales every term containing it by λ, and summing the scaled terms reproduces the homogeneity identity directly from the distributive and associative laws of ordinary multiplication.

Failure of Homogeneity as a Diagnostic

If a proposed component formula involves a coordinate raised to any power other than one, or combined nonlinearly with itself, scaling that coordinate by λ scales the corresponding term by a power of λ different from one, violating homogeneity; such a formula cannot correspond to a tensor, since tensors are defined precisely as maps satisfying homogeneity, together with additivity, in every slot.

Homogeneity Under Partial Evaluation

Preservation in the Remaining Open Slots

When some slots of T are fixed through partial evaluation, the resulting tensor remaining slot map continues to satisfy homogeneity in each of its still-open slots, since homogeneity in a slot depends only on the behavior of T in that slot with the rest held fixed, unaffected by whatever specific values occupy the fixed slots.

Homogeneity in the Fixed Arguments

The reduced arity result produced by partial evaluation also depends homogeneously on each fixed argument, which is the homogeneous half of tensor partial evaluation linearity; scaling a fixed argument by λ scales the entire resulting reduced arity result, viewed as a tensor, by the same λ.

Diagrammatic Summary

The diagram illustrates homogeneity by showing that scaling a single argument by λ produces an output scaled by exactly that same factor λ, with all other arguments unchanged.