4.14.3 Tensor General Slot Homogeneity

Tensor General Slot Homogeneity is the uniform statement of the tensor multilinear homogeneity property across an arbitrary, unspecified slot of a tensor, expressing homogeneity as a single condition indexed by slot position rather than as a separate identity written out for each individual slot such as the first or the second. It packages the pattern observed in tensor first slot homogeneity and tensor second slot homogeneity, and in every other individual slot, into one general schema, making explicit that homogeneity is required uniformly across all p + q slots of a tensor, with no slot exempted or treated differently in kind.

The General Schema

Homogeneity Indexed by an Arbitrary Slot

For a type (p, q) tensor T on a vector space V over a field F, let k range over any of the p + q slots. General slot homogeneity states that for any admissible argument v for slot k, any scalar λ ∈ F, and any fixed choice of every other argument,

with the notation understood to place λv in slot k specifically, while every other slot, indicated by the placeholders ⋯, remains fixed and identical on both sides. The index k ranges over every one of the p + q positions, so general slot homogeneity is a family of p + q individual statements, one for each value of k, unified under a single schema.

Quantifying Over All Slots at Once

General slot homogeneity can be stated compactly as the requirement that, for every k from 1 to p + q, the map obtained by fixing all slots except the k-th is homogeneous in its one remaining argument; this universally quantified form is what distinguishes the general property from any single slot-specific instance such as first slot homogeneity, which fixes k = 1 outright.

Relation to the Specific Slot Instances

Recovering First and Second Slot Homogeneity as Special Cases

Setting k = 1 in the general schema recovers exactly the identity described by tensor first slot homogeneity, and setting k = 2 recovers exactly the identity described by tensor second slot homogeneity; every specific slot instance of homogeneity is obtained from the general schema by substituting a particular value for k, so the general property does not add new mathematical content beyond what is already present in the collection of all individual slot instances, but it does express their common pattern as a single statement.

No Slot Is Structurally Distinguished

The general schema makes explicit that homogeneity does not privilege any particular slot: whatever scaling behavior holds for the first or second slot must equally hold for the third, the last, or any slot in between, with the only difference between slots being whether they are contravariant or covariant, determining the kind of argument, vector or covector, that homogeneity is stated for at that position.

Why the General Form Is Needed

Tensors of Arbitrarily High Rank

For a tensor of low rank, such as a bilinear form with only two slots, it is practical to write out first slot homogeneity and second slot homogeneity as two separate identities; for a tensor of rank p + q much larger than two, writing out a separate homogeneity identity for every individual slot becomes unwieldy, and the general schema, indexed by an arbitrary k, is the practical way to state the full requirement without enumerating every slot by name.

Uniformity Required by the Definition of Multilinearity

The definition of a multilinear map requires homogeneity, together with additivity, in every slot without exception; general slot homogeneity is the precise formal expression of this "every slot without exception" requirement, ensuring that no proof or verification of multilinearity can be considered complete while omitting any slot from consideration.

Consequences of the General Property

Scaling Across Multiple Slots Simultaneously

Applying general slot homogeneity to several slots in succession, each scaled by its own scalar, shows that the total output scales by the product of all the individual scalars; this repeated application, licensed by general slot homogeneity holding at every slot, is the basis for the result described in tensor multilinear evaluation result concerning the simultaneous scaling of every argument at once.

Foundation for the Component Evaluation Formula

The tensor multilinear component evaluation formula relies on general slot homogeneity holding at every one of the p + q slots simultaneously; each term in the formula involves the coordinates of every argument multiplied together with the corresponding component of T exactly once, a pattern that is only consistent with the tensor's output if homogeneity holds uniformly across every one of those slots.

Diagrammatic Summary

The diagram shows every slot of the tensor, from the first to the last, subject to the same homogeneity pattern, with the dashed box marking an arbitrary slot k to emphasize that the property holds uniformly no matter which slot is singled out.