14.19.2 Tensor Map Product Component Transformation

Tensor Map Product Component Transformation describes how tensor components transform under a map, linking algebraic structure to coordinate changes in tensor spaces.

Tensor Map Product Component Transformation is the index-level transformation law describing how each individual entry of the component array of $f \otimes g$ changes under a change of basis, expressed through the upper index, associated with the codomain, transforming contravariantly and the lower index, associated with the domain, transforming covariantly, with the combined object inheriting one covariant and one contravariant transformation from each of the two factors $f$ and $g$ .

Covariant and Contravariant Index Behavior

Upper Index Transforms Contravariantly

A linear map $f : V \to V^{'}$ has component $F_{i}^{a}$ carrying a lower index $i$ , referring to the domain basis, and an upper index $a$ , referring to the codomain basis. Under a change of basis on the codomain given by an invertible matrix $P^{'}$ , the upper index transforms contravariantly,

{\tilde{F}}_{i}^{c} = \sum_{a} {(P^{'})}_{a}^{c} F_{i}^{a}

meaning the new coefficients are obtained by contracting with $P^{'}$ directly, matching how the components of a vector, rather than of a basis, transform.

Lower Index Transforms Covariantly

Under a change of basis on the domain given by an invertible matrix $P$ , the lower index transforms covariantly, meaning it is contracted with the inverse of $P$ rather than with $P$ itself,

{\tilde{F}}_{k}^{a} = \sum_{i} F_{i}^{a} {(P^{- 1})}_{k}^{i}

which is exactly the transformation appropriate to an index that labels a basis vector directly rather than a coordinate relative to that basis.

Combined Transformation for the Tensor Product

Four Independent Contractions

Combining both types of index behavior for both $f$ and $g$ , the full component transformation for $f \otimes g$ is

{(\tilde{f} \otimes \tilde{g})}^{c d}_{k l} = \sum_{i} \sum_{j} \sum_{a} \sum_{b} {(P^{'})}_{a}^{c} {(Q^{'})}_{b}^{d} F_{i}^{a} G_{j}^{b} {(P^{- 1})}_{k}^{i} {(Q^{- 1})}_{l}^{j}

with four independent sums, one contravariant contraction for each output index and one covariant contraction for each input index, all performed on the two original factor coefficients simultaneously.

Reduction to Matrix Form

Regrouping this expression by which sums belong to $f$ and which belong to $g$ shows it factors into the product of the ordinary component transformation of $F$ alone and the ordinary component transformation of $G$ alone, which is the component-level restatement of the matrix identity $\tilde{F} \otimes \tilde{G} = (P^{'} \otimes Q^{'}) (F \otimes G) {(P \otimes Q)}^{- 1}$ .

Interpretation of the Two Index Types

Why the Distinction Matters

The reason the upper and lower indices transform with $P^{'}$ and $P^{- 1}$ respectively, rather than both with the same matrix, is that a component array mixes two different roles: the lower index picks out which domain basis vector is being mapped, and re-expressing that basis vector in a new basis requires the inverse change of basis matrix, while the upper index picks out how much of a codomain basis vector appears in the result, and re-expressing the result in a new codomain basis requires the change of basis matrix directly, not its inverse.

Behavior of a Purely Contravariant or Purely Covariant Slot

If the codomain bases are held fixed, the transformation reduces to purely covariant behavior in the lower indices; if the domain bases are held fixed, it reduces to purely contravariant behavior in the upper indices, and the general case interpolates between these two extremes by allowing both to act at once.

Consistency With Invariant Quantities

Fully Contracted Scalars Are Unaffected

Whenever every upper index of the component array is fully contracted against a matching lower index, for instance in forming a trace of an endomorphism obtained by setting $V = V^{'}$ and $W = W^{'}$ , the factors of $P^{'}$ and $P^{- 1}$ cancel against one another, leaving the resulting scalar completely independent of the basis change, which is the component-level reason traces and other fully contracted invariants of $f \otimes g$ do not depend on the choice of basis.