4.14.1 Tensor First Slot Homogeneity

Tensor First Slot Homogeneity is the specific instance of the tensor multilinear homogeneity property applied to the first argument position of a multilinear map, requiring that scaling the first argument by a constant, while every other argument is held fixed, produces an output scaled by exactly that same constant. It is the natural counterpart to tensor first slot additivity, and together the two conditions establish that the first slot of a tensor behaves as an ordinary linear map when every other slot is held fixed.

Statement for the First Slot

The Defining Identity

For a type (p, q) tensor T on a vector space V over a field F, first slot homogeneity states that for any admissible argument u in the first position, any scalar λ ∈ F, and any fixed choice of every remaining argument,

where x_2, ... denotes every argument from the second slot onward, held fixed and identical on both sides. Only the first slot is scaled, by λ on the left, while the entire output on the right is scaled by that same λ.

The Kind of Object Filling the First Slot

As with first slot additivity, whether the first slot is contravariant or covariant is fixed by the type of T; first slot homogeneity is stated using whichever kind of argument, covector or vector, that slot expects, and the scalar λ acts on that argument by the ordinary scalar multiplication defined on V or V* respectively.

Illustration in the Bilinear Case

The Simplest Setting: Two Arguments

For a type (0, 2) tensor B, a bilinear form taking two vector arguments, first slot homogeneity reads

with the second argument y held fixed throughout, showing that B responds proportionally to any rescaling of its first argument, for every fixed choice of the second.

Contrast with Second Slot Homogeneity

Second slot homogeneity for the same bilinear form B instead requires B(x, λy) = λ B(x, y), with the first argument fixed; first slot homogeneity and second slot homogeneity are logically independent conditions, and a bilinear form must satisfy both, since multilinearity requires homogeneity in every slot simultaneously, not merely in one.

First Slot Homogeneity and Symmetry Relations

Transfer Under Symmetric Tensors

If a tensor is symmetric in its first two slots, so that swapping the first and second arguments leaves the output unchanged, first slot homogeneity and second slot homogeneity become logically equivalent statements about that tensor, since the symmetry relation directly translates the scaling behavior in one slot into the scaling behavior in the other.

Transfer Under Antisymmetric Tensors

If a tensor is antisymmetric in its first two slots, so that swapping the first and second arguments negates the output, first slot homogeneity again implies second slot homogeneity, and vice versa, since scaling one side of the antisymmetry relation and comparing coefficients reproduces the homogeneity identity for the other slot.

Consequences Restricted to the First Slot

Vanishing When the First Argument Is Zero

Setting λ = 0 in first slot homogeneity shows that T(0, x_2, ...) = 0 for any fixed choice of the remaining arguments, meaning the tensor automatically vanishes whenever its first argument is the zero vector or zero covector, regardless of what values occupy the other slots.

First Slot Homogeneity Alone Is Insufficient for Multilinearity

Establishing that a candidate map satisfies first slot homogeneity says nothing, by itself, about its behavior in the second, third, or any later slot; a map could be homogeneous in its first argument while failing homogeneity in a later argument, and such a map would not be classified as fully multilinear despite satisfying first slot homogeneity in isolation.

Its Place in a Full Multilinearity Check

Paired with First Slot Additivity

Confirming that the first slot of a tensor behaves linearly requires checking both first slot additivity and first slot homogeneity together; neither condition alone establishes linearity in the first slot, since a map could satisfy one without the other, for instance if it involves an absolute value or another operation that respects sums but not scalar multiples, or vice versa.

One Step Among Many

First slot homogeneity is simply the first of p + q homogeneity checks needed to establish full multilinearity of a tensor, one for each slot; it must be supplemented by identical homogeneity checks, alongside additivity checks, for every remaining slot before multilinearity can be concluded.

Diagrammatic Summary

The diagram illustrates first slot homogeneity by showing the second argument y held fixed while scaling the first argument by λ produces an output scaled by that same λ.