13.2.3 Tensor Partial Contraction Area

Tensor Partial Contraction Area involves selectively contracting tensor indices to reduce rank while maintaining structural integrity in multilinear algebra.

Tensor Partial Contraction Area is the domain within tensor contraction areas concerned with contractions that reduce a tensor's order without eliminating every index, leaving at least one free index in the resulting tensor and distinguishing this case from contraction carried through to a scalar.

Defining the Partial Contraction Case

Reduction Without Elimination

A partial contraction area applies whenever the number of independent contractions performed on a tensor is fewer than half the tensor's original order, so that some contravariant or covariant indices remain unpaired and persist as free indices in the result.

T_{i k}^{i l} = \sum_{i = 1}^{n} T_{i k}^{i l}

In this expression, the indices $k$ and $l$ remain free after the contraction over $i$ , placing the result within the partial contraction area.

Nonzero Residual Order

The defining numerical signature of the partial contraction area is a resulting order strictly greater than zero, reflecting the presence of at least one uncontracted index remaining after the operation is complete.

order (result) > 0

Behavior of Free Indices Under Partial Contraction

Preservation of Variance and Dimension

Every free index remaining after a partial contraction retains the same variance, contravariant or covariant, and the same dimension it possessed on the original tensor, since contraction affects only the indices it explicitly pairs and leaves all others unaltered.

Transformation Behavior of the Remaining Tensor

A tensor produced by partial contraction transforms under a change of basis according to the standard transformation law associated with its remaining free indices, following the same rule that would apply to a tensor of that reduced order constructed by any other means.

{\tilde{T}}^{k} = \frac{\partial {\tilde{x}}^{k}}{\partial x^{l}} T^{l}

Instances Within the Partial Contraction Area

Reducing Order by Two Without Reaching Zero

A tensor of high order contracted on exactly one pair of indices, while retaining several other indices uncontracted, exemplifies the simplest instance of partial contraction, reducing order by exactly two while leaving the remainder of the tensor's structure intact.

Contracting an Operator Against a Vector

The action of a linear operator on a vector, expressed by contracting one index of a rank-two tensor against the single index of a vector, exemplifies a partial contraction that produces a vector rather than a scalar, since one index of the operator remains free in the result.

w^{i} = A_{j}^{i} v^{j}

Distinction from Full Contraction

Retained Versus Eliminated Structure

Where full contraction, associated with tensor scalar result scope, removes every index and yields a basis-independent value, partial contraction retains a genuine tensor with structure that still depends on the choice of basis for its explicit numerical description, even though the tensor itself remains a well-defined invariant object.

Verification Differences

Invariance verification applied within the partial contraction area confirms that the result transforms according to the standard tensor transformation law for its remaining indices, a comparatively more involved check than the simple numerical equality across bases required when verifying a fully contracted scalar.

Relationship to Tensor Operation Notation

The partial contraction area is signaled in tensor operation notation by an expression containing one or more repeated index symbols alongside one or more index symbols that appear only once, with the singly appearing symbols identifying the free indices that remain in the resulting tensor once the indicated contractions have been carried out.