14.19.5 Tensor Map Product Tensorial Consistency

Tensor Map Product Tensorial Consistency ensures compatibility and structure preservation in tensor algebra operations across mapped spaces.

Tensor Map Product Tensorial Consistency is the property that the component array of $f \otimes g$ , under the transformation law established for it, actually behaves as a genuine tensor of mixed type rather than merely as an arbitrary array of numbers that happens to be attached to a basis, and it is what justifies referring to the entries of $F_{i}^{a} G_{j}^{b}$ collectively as a tensor rather than as a mere table of coefficients.

What Tensorial Consistency Requires

The General Definition Being Checked

An array of numbers indexed by some upper and lower indices is called a tensor of a given type, relative to a family of bases, only if its transformation under a change of basis is governed entirely by the change of basis matrix and its inverse, applied once for every index, with no additional dependence on the array's own numerical values beyond the transformation law itself. Tensorial consistency is the verification that the component array of $f \otimes g$ satisfies exactly this requirement.

The Type Being Claimed

The component $F_{i}^{a}$ of a single linear map is a tensor of type $(1, 1)$ , with one contravariant index $a$ and one covariant index $i$ . The claim of tensorial consistency for the tensor product is that the combined array $F_{i}^{a} G_{j}^{b}$ , viewed as a single object with two contravariant indices $a, b$ and two covariant indices $i, j$ , is a tensor of type $(2, 2)$ .

Verifying Consistency Directly

The Transformation Already Established

The component transformation derived from the basis change response shows that under simultaneous changes of basis, each upper index of the combined array is contracted with a copy of the relevant change of basis matrix, and each lower index is contracted with a copy of the corresponding inverse, exactly matching the general pattern required of a type $(2, 2)$ tensor, with no extra terms and no dependence on anything besides the four matrices $P, P^{'}, Q, Q^{'}$ and the original components.

Transitivity Under Composed Basis Changes

A further requirement of tensorial consistency is that applying two successive changes of basis in sequence gives the same result as applying their composite directly. Writing the two changes as $P_{1}, P_{2}$ for the $V$ -side, the component transformation satisfies

\sum_{k} {(P_{2})}_{k}^{c} (\sum_{i} {(P_{1})}_{i}^{k} F_{i}^{a}) = \sum_{i} {(P_{2} P_{1})}_{i}^{c} F_{i}^{a}

matching the single-step transformation by the composite matrix $P_{2} P_{1}$ , and the identical property holds separately on the $W$ -side and for the two Kronecker factors together, since Kronecker products of composites equal composites of Kronecker products.

Consistency With the Identity Change

Taking the change of basis matrices to be the identity on all four spaces must return the original component array unchanged, which follows immediately since every contraction with an identity matrix leaves an array untouched; this is the base case anchoring the transitivity check above.

Why This Matters Beyond Bookkeeping

Distinguishing Tensors From Arbitrary Arrays

Not every doubly or quadruply indexed array of numbers attached to a choice of basis is a tensor; an array might instead represent, for instance, the coefficients of a particular numerical algorithm that depend on the basis in some way not reducible to contraction with change of basis matrices. Tensorial consistency is precisely the check that rules out this possibility for the tensor product of maps: its component array transforms exactly as a tensor should, and no more elaborately.

Basis for Further Constructions

Because the component array of $f \otimes g$ passes this consistency check, it can be combined with other tensors, contracted, symmetrized, or otherwise manipulated using the standard operations of tensor calculus, with the guarantee that the results of those operations will themselves transform correctly under further changes of basis, a guarantee that would fail if the underlying array were not tensorially consistent to begin with.