13.6.4 Tensor Full Contraction Dimension Compatibility

Tensor Full Contraction Dimension Compatibility ensures valid dimensionality during full index contraction in tensor algebra.

Tensor Full Contraction Dimension Compatibility is the condition that every member of the all slot pairing underlying a full contraction operation must independently satisfy the dimension requirement, ensuring that each of the several contracted pairs links two slots ranging over vector spaces of equal dimension.

Statement of the Compatibility Condition

Equal Dimension Required for Every Pair

For a full contraction operation to be valid, each pair within the associated all slot pairing must consist of a contravariant slot and a covariant slot whose underlying vector spaces share the same dimension, with this requirement applying separately and independently to every pair in the pairing.

dim (V_{i}) = dim (V_{i}), dim (V_{j}) = dim (V_{j})

An Extension of the Single Pair Requirement

This compatibility condition extends the dimension requirement already established for a single contraction slot pair to every member of a slot pair set simultaneously, reflecting the fact that full contraction is composed of several elementary contractions each subject to that same underlying requirement.

Why Compatibility Is Guaranteed in Common Cases

Automatic Satisfaction Within a Single Tensor Built from One Space

When the tensor undergoing full contraction is built entirely from a single vector space and its dual, every contravariant and every covariant index shares that same common dimension, so dimension compatibility across every pair of the all slot pairing is automatically satisfied without further verification.

T_{i j}^{i j}

A Condition Requiring Explicit Confirmation for Products of Tensors

When full contraction is applied to a product formed from several separate tensors, each potentially built from a different vector space, dimension compatibility must be explicitly confirmed for every pair, since indices contributed by different factors are not guaranteed in advance to share a common dimension.

A_{i} B^{i}

Consequence of a Failure in Compatibility

Absence of a Well-Defined Summation Range

If any single pair within a proposed all slot pairing fails to satisfy dimension compatibility, the summation associated with that particular pair lacks a consistent range over which to proceed, since the two slots being linked would disagree on how many values the shared summation index may take.

Invalidity of the Entire Full Contraction

Because a full contraction operation requires every pair within its all slot pairing to be individually valid, a dimension incompatibility in even one pair renders the entire operation undefined, regardless of whether the remaining pairs would otherwise satisfy the requirement on their own.

Verification of Compatibility Across All Pairs

Pairwise Checking Within Input Verification

Dimension compatibility for a full contraction is confirmed during input verification by checking each member of the all slot pairing separately, rather than through a single aggregate check, since satisfying the requirement for one pair provides no information about whether another pair in the same pairing also satisfies it.

Necessity of Checking Every Pair Before Computation

Because the full contraction operation only produces a well-defined scalar when every pair is valid, confirming dimension compatibility across the entire all slot pairing must be completed before index summation begins for any of the individual pairs.

Relationship to Tensor Operation Notation

Dimension compatibility across a full contraction is not directly visible within tensor operation notation, since the notation records only the labels and placement of indices rather than their associated dimensions, but the requirement governs which notated expressions, featuring several matched repeated index pairs, correspond to a full contraction that is genuinely well defined once the dimensions of every pair have been examined.