13.4.5 Tensor Single Contraction Result Structure

Tensor single contraction result structure reveals how tensor contraction simplifies relationships and exposes key algebraic properties.

Tensor Single Contraction Result Structure is the specific arrangement of free indices, variance, and dimension characterizing the tensor produced by a single index contraction operation, describing the shape of the output that follows from removing exactly one contravariant and one covariant slot from the input.

Components of the Result Structure

The List of Remaining Free Indices

The result structure begins with the ordered list of indices carried by the input tensor other than the two forming the contracted slot pair, with the relative order of these remaining indices typically preserved from their arrangement on the original input.

T_{i k}^{i l} \to S_{k}^{l}

Variance Assigned to Each Remaining Index

Each free index in the result structure retains exactly the variance, contravariant or covariant, that it held as part of the input tensor, since the contraction operation acts only upon the selected slot pair and leaves the character of every other slot unaffected.

Dimension Assigned to Each Remaining Index

Each free index in the result structure retains the dimension of the vector space it ranged over on the input tensor, so the result structure inherits, unchanged, the dimensions associated with every index not involved in the contracted pair.

The Order of the Result Structure

A Fixed Reduction from the Input Order

The total number of indices comprising the result structure equals the order of the input tensor reduced by exactly two, reflecting the removal of the one contravariant and one covariant index consumed by the operation.

order (S) = order (T) - 2

The Degenerate Case of Zero Remaining Indices

When the input tensor's order is exactly two, the result structure contains no free indices at all, reducing to the structure of a scalar, a special case in which the result carries no index arrangement to describe beyond its single numerical value.

Structural Independence from Numerical Values

Structure Fixed Before Computation

The result structure, comprising the identity, variance, and dimension of the remaining free indices, is fully determined once the input tensor's structure and the selected slot pair are known, independent of the specific numerical values that index summation will later assign to each component.

Distinguishing Structure from Computed Content

Where tensor contraction index summation concerns the specific numerical values populating the result, result structure concerns only the framework of index positions those values will occupy, so structure and computed content together, but separately, determine the complete result tensor.

Result Structure Under Multiple Contractions

Cumulative Effect on Structure

When several single index contraction operations are applied together, the result structure reflects the combined removal of every contracted slot pair, retaining only those indices from the original tensor untouched by any of the applied pairings.

T_{i j}^{i j} \to c

Consistency Regardless of Contraction Order

Because each contraction's effect on structure depends only on which slots it removes, applying several single index contraction operations in different sequences yields the same final result structure, provided the same set of slot pairs is ultimately contracted.

Relationship to Tensor Operation Notation

Result structure is read directly from tensor operation notation by identifying which index symbols remain unrepeated after accounting for the contracted pair, since these singly appearing symbols, together with their upper or lower placement, constitute the complete notated description of the result structure produced by the operation.