4.6.2 Tensor Slotwise Homogeneity Property

Tensor Slotwise Homogeneity Property is the half of the multilinearity condition asserting that scaling the argument in a single slot by a scalar, while holding every other slot fixed, scales the output of a tensor's defining map by exactly that same scalar. It is the "scale-preserving" component of slotwise linearity, distinct from additivity, and it is what pins down the precise numerical relationship between an argument's magnitude and the tensor's output within any one slot.

Formal Statement

The Homogeneity Condition

For a multilinear map $T : V_1 \times \cdots \times V_k \to F$, slotwise homogeneity requires that for every slot index $i$, every vector $u \in V_i$, and every scalar $c \in F$,

with the arguments in every other slot fixed and identical on both sides. This condition must hold independently at each of the $k$ slots, with no interaction assumed between the scaling of one slot and the values held in any other.

Distinction from Homogeneity of the Whole Map

Slotwise homogeneity concerns scaling only one argument at a time. It should not be confused with the degree-$k$ homogeneity of the tensor viewed as a function of the entire tuple: scaling every argument simultaneously by the same scalar $c$ produces an output scaled by $c^{k}$, obtained by applying slotwise homogeneity once in each of the $k$ slots in succession, not by a single global homogeneity condition.

Consequences of Homogeneity Alone

Vanishing at the Zero Vector

Setting $c = 0$ in the homogeneity condition immediately gives $T(\ldots, 0, \ldots) = 0$ for any slot, independent of the values in the other slots. This is the most elementary consequence of slotwise homogeneity and holds regardless of additivity.

Sign Reversal

Setting $c = -1$ shows that reversing the direction of the vector in any one slot reverses the sign of the tensor's output in that slot:

Pulling Coefficients Outside Component Sums

Homogeneity is what allows the scalar coefficient of a basis expansion to be factored entirely outside the tensor evaluation, once additivity has already split a sum into individual terms: each basis vector's coefficient can be pulled in front of $T$, leaving a bare basis vector inside the slot, which is precisely the operation used to reduce any tensor evaluation to a weighted sum of components.

Role Within the Broader Multilinearity Property

Complementary to Additivity

Slotwise homogeneity and slotwise additivity are independent but jointly necessary conditions: a map satisfying only homogeneity in a slot, without additivity, would scale correctly but might not behave predictably under sums of vectors in that slot, breaking the finite-component expansion that both conditions together guarantee.

Testing Homogeneity in Practice

To verify slotwise homogeneity for a candidate multilinear map, it suffices to check the scaling identity for arbitrary scalars and an arbitrary vector in one slot at a time, with all other slots held at arbitrary but fixed values, repeating the check independently for each of the $k$ slots; a single violation in any slot disqualifies the map from being multilinear, regardless of how well the map behaves under sums.

Summary of Key Points

• Slotwise homogeneity requires that scaling the argument in one slot by $c$, with all other slots fixed, scales the tensor's output by exactly $c$.
• It is logically independent from slotwise additivity, though both together constitute full slotwise linearity.
• Applying it once per slot to a simultaneous scaling of every argument produces the degree-$k$ homogeneity characteristic of arity-$k$ multilinear maps.
• Setting $c = 0$ or $c = -1$ yields the elementary vanishing and sign-reversal properties of tensors.
• Homogeneity is what allows scalar coefficients from a basis expansion to be factored entirely outside a tensor evaluation.