4.14.5 Tensor Homogeneity Component Form

Tensor Homogeneity Component Form is the expression of the tensor multilinear homogeneity property directly in terms of the numerical coordinates of an argument once a basis has been fixed, showing that homogeneity in a slot corresponds exactly to the ordinary scalar multiplication of that argument's coordinate array. It translates the abstract, basis-independent homogeneity identity into a concrete statement about how scalar multiples of coordinate arrays behave when substituted into the component evaluation formula for a tensor.

Setting Up the Component Form

Coordinates of a Scalar Multiple

Let V be a finite-dimensional vector space with basis e_1, ..., e_n. If a vector v has coordinates v^j relative to this basis, the scaled vector λv, for a scalar λ, has coordinates equal to λ times each coordinate of v:

This is simply the definition of scalar multiplication of a vector expressed in coordinates, and it is the starting point for translating homogeneity into a statement purely about numbers.

Substituting the Scaled Coordinates

For a type (p, q) tensor T with components T^{i_1 ... i_p}_{j_1 ... j_q}, substituting the coordinates of λv into the l-th covariant slot of the component evaluation formula gives, for that slot's index j_l, the coordinate λv^{j_l} in place of a single unscaled coordinate.

The Homogeneity Identity in Coordinates

Factoring the Scalar Through the Component Formula

Because ordinary multiplication is associative and commutative, the term T^{i_1 ... i_p}_{j_1 ... j_l ... j_q}(λv^{j_l}), appearing inside the summation over j_l, can be rewritten with the scalar λ factored to the front:

Carrying this factoring through the entire summation over every index, since λ does not depend on any of the other summation indices and can be pulled outside every one of them, reproduces exactly the homogeneity identity T(..., λv, ...) = λ T(..., v, ...), now derived purely from the associative and distributive properties of ordinary arithmetic applied to the component sum.

Homogeneity as a Property of Linear-in-Coordinate Formulas

More generally, the component evaluation formula exhibits homogeneity in slot l precisely because the formula involves the coordinate v_l^{j_l} associated with that slot to exactly the first power in every term: any formula in which that coordinate appeared to a different power, or nested inside a nonlinear function, would not permit λ to be factored out cleanly and would therefore fail homogeneity in that slot.

Verifying Homogeneity by Inspecting Components

The Linear-in-Coordinates Criterion

Given an explicit component formula for a candidate multilinear map, homogeneity in a given slot can be verified directly by checking that the formula involves the coordinate associated with that slot exactly to the first power in every term, since only in that case does scaling the coordinate by λ scale every term, and hence the entire sum, by exactly λ.

Example of a Formula Satisfying Homogeneity

The bilinear form component formula ∑_{i,j} T_{ij} v^i w^j satisfies homogeneity in its second argument because w^j appears only to the first power; replacing w with λw scales every term, and hence the whole sum, by λ, exactly reproducing the homogeneity identity.

Example of a Formula Failing Homogeneity

A formula such as ∑_i T_i (v^i)^2, in which the coordinate appears squared, fails homogeneity, since scaling v by λ scales this expression by λ^2 rather than by λ; such a formula does not describe a multilinear map and therefore cannot correspond to any tensor.

Homogeneity Component Form and Basis Independence

The Same Homogeneity Holds in Every Basis

Although the component form of homogeneity is expressed relative to a specific basis, the underlying homogeneity property of T itself does not depend on that basis; expressing the same tensor in a different basis changes the numerical values of the components and coordinates involved, but the factoring argument that establishes homogeneity continues to hold identically, since it follows from the same underlying linear-in-coordinate structure in any basis.

Consistency with Basis-Free Homogeneity

The component form of homogeneity is not a separate or weaker version of the basis-free homogeneity identity; it is the same identity, made concrete by substituting specific numerical coordinates, and the two formulations agree exactly once the coordinate expansion of the abstract vector is substituted into the abstract identity.

Diagrammatic Summary

The diagram shows how the scaled coordinate λv^j, substituted into the component evaluation formula, factors cleanly out of the sum to yield exactly λ times the unscaled component evaluation.