10.10.3 Tensor Matrix Component Mixed Index Update

Exploring how mixed index tensors update matrix components through algebraic transformations in tensor algebra.

Tensor Matrix Component Mixed Index Update is the operation of relabeling and recomputing both the upper and the lower index of a mixed rank-two tensor's component array simultaneously when a change of basis is applied, attaching a prime to each index while contracting the old components against the inverse matrix for the upper index and the forward matrix for the lower index in a single combined computation. It extends the single-index update used for vectors and covectors to an object with two indices of opposite type, requiring both updates to be carried out together so that neither index is left inconsistent with the other.

What the Update Consists Of

Attaching Primes to Both Indices

When a mixed tensor's components are transformed to a new basis, both the upper index and the lower index acquire a prime, marking every subsequent reference to that component as belonging to the new basis rather than the old one.

M_{j}^{i} \to M_{j}^{i}'

Recomputing the Value Through Two Simultaneous Contractions

Alongside the relabeling, the index update requires the actual numerical value of the new component to be recomputed by contracting the old components with the inverse matrix for the upper index and the forward matrix for the lower index, both within the same expression.

M_{j}^{i}' = {(A^{- 1})}_{k}^{i} A_{j}^{l} M_{l}^{k}

Necessity of Updating Both Indices Together

Partial Updates Break Consistency

Updating only the upper index while leaving the lower index unprimed, or vice versa, would produce a hybrid expression describing neither the old basis nor the new basis consistently, since one index would refer to the new representation while the other still referred to the old one.

Distinct Matrix Factor for Each Index

Because the upper index requires the inverse matrix and the lower index requires the forward matrix, the mixed index update cannot be performed by applying a single matrix factor to the whole component array; the two contractions must be carried out separately, each acting on its own index, before being combined into the fully updated component.

Relationship to Single-Index Updates

Composition of the Vector and Covector Index Updates

The mixed index update can be understood as the simultaneous application of the vector component index update to the upper index and the covector component index update to the lower index, performed within a single tensor rather than on two separate objects.

Consistency With the Similarity Transformation

Expressed in matrix notation, the mixed index update corresponds exactly to forming the similarity transformation of the tensor matrix, with the left multiplication by the inverse matrix updating the upper index and the right multiplication by the forward matrix updating the lower index.

M' = A^{- 1} M A

Consequences for Notation Discipline

Verifying Both Contractions Independently

Because two distinct contractions are involved, a formula can be checked for correctness by verifying each index update separately: confirming that the upper index has indeed been contracted with the inverse matrix and that the lower index has indeed been contracted with the forward matrix, independent of one another.

Extension to Tensors With More Indices

The same principle of updating every index with its own appropriate matrix factor, simultaneously and consistently, extends directly to tensors carrying any combination of upper and lower indices beyond the single pair considered in the rank-two mixed case.

Schematic Representation

The diagram shows both indices of the mixed tensor transforming together, one via the inverse matrix and the other via the forward matrix, the two simultaneous contractions that together constitute the mixed index update.