16.9.3 Tensor Alternating Multilinear Component Constraint
The Tensor Alternating Multilinear Component Constraint enforces antisymmetry in tensor components, ensuring multilinearity and alternating properties across indices.
Tensor Alternating Multilinear Component Constraint is the requirement, distinct from the purely combinatorial sign-swap constraint on index labels, that a multilinear form's components must be mutually consistent with the linearity condition itself — meaning the value predicted for an argument built as a linear combination of basis vectors must match the value obtained by expanding that combination through the stored components, with no discrepancy permitted.
The Constraint Arising from Linearity
Statement of the Constraint
For an alternating multilinear form T with stored components T_{i₁...iₖ} relative to a basis, and for arbitrary vectors v₁,...,vₖ expressed as linear combinations vⱼ = Σᵢ aᵢⱼeᵢ, the multilinear component constraint requires:
This constraint requires the component array, together with the linearity-driven expansion formula on the right, to reproduce the actual value of T on every possible combination of vectors, not merely on basis vectors.
Distinguishing This from the Sign-Swap Constraint
The sign-swap constraint (relating T_{...i...j...} to T_{...j...i...}) governs relationships purely among components, holding the index structure fixed. The multilinear component constraint instead governs the relationship between the entire component array and the evaluation of T on non-basis vectors, a fundamentally different type of consistency requirement layered on top of the sign-swap constraint rather than replacing it.
Why Both Constraints Are Needed Together
Neither Constraint Alone Suffices
Satisfying only the sign-swap constraint without the multilinear expansion constraint would allow a component array that is internally sign-consistent but fails to represent any actual multilinear function, since nothing would tie the stored numbers to a well-defined rule for evaluating combinations. Conversely, satisfying only multilinearity without the sign-swap constraint permits general (non-alternating) tensors, which are multilinear but do not have the antisymmetric relationships among components.
The Combined Constraint Characterizes Alternating Tensors Exactly
Together, the two constraints — sign-swap consistency among components, and multilinear consistency between components and arbitrary-vector evaluation — exactly characterize what it means for a component array to represent a genuine alternating multilinear form; no additional condition is needed beyond these two.
Consequence: Redundant Encoding Detection
Verifying a Candidate Array
Given any proposed array of numbers T_{i₁...iₖ}, the multilinear component constraint provides a concrete test: pick several combinations of basis vectors, evaluate the right-hand expansion formula, and confirm the results are consistent with any independently known values of T on those combinations. Failure at any single combination certifies the array as inconsistent with genuine multilinearity.
Determinant Expansion as an Instance
The Leibniz formula for a determinant is the specific instance of this constraint applied to the top-rank alternating form, where the "components" are the single scalar T_{1,...,n} and the expansion formula reduces to the familiar signed sum over permutations of matrix entries:
directly demonstrating the general constraint specialized to k = n.