13.2.2 Tensor Multiple Contraction Area

Tensor Multiple Contraction Area explores how multiple tensor indices are contracted, revealing deep algebraic structures in multilinear algebra and physical applications.

Tensor Multiple Contraction Area is the domain within tensor contraction areas concerned with expressions in which two or more independent contractions are applied simultaneously within the same tensor or product of tensors, each summing over its own distinct pair of indices.

Defining the Multiple Contraction Case

Several Independently Paired Index Sets

A multiple contraction area applies when an expression contains more than one repeated index symbol, with each repeated symbol marking a separate pair of one contravariant and one covariant index summed independently of the others.

T_{i j}^{i j} = \sum_{i = 1}^{n} \sum_{j = 1}^{n} T_{i j}^{i j}

Independence of the Paired Sets

Each contraction within this area is scoped to its own repeated symbol, with the summation over one pair of indices proceeding without regard to the summation over another pair, so that the multiple contractions act independently even though they are carried out within a single combined expression.

Order Change Within the Multiple Contraction Area

Cumulative Reduction Across All Contractions

The order of the result decreases by two for every independent contraction present in the expression, so that an expression containing several contractions reduces the total order by twice the number of contractions applied.

order (result) = order (T) - 2 k

Here $k$ denotes the number of independent contractions present in the expression.

Full Reduction to a Scalar as a Special Case

When the number of independent contractions equals half the original order of the tensor, every index is consumed and the multiple contraction area produces a scalar, generalizing the endpoint reached by full contraction to expressions involving several simultaneous pairings.

Applying Contractions in a Product of Tensors

Independent Pairings Across Separate Factors

When multiple contractions act on indices drawn from a product of several tensors, each pairing may connect indices belonging to different factors, so that the multiple contraction area encompasses cases where the several summations link the factors of the product together in more than one place.

A_{i} B_{j}^{i} C^{j}

Order of Application Among Independent Contractions

Because the contractions within this area are independent of one another, the order in which the individual summations are carried out does not affect the final result, allowing them to be evaluated in any sequence or, in principle, simultaneously.

Verification Within a Multiple Contraction Area

Checking Each Pairing Separately

Verification within the multiple contraction area requires applying the checks for slot existence, opposite variance, and dimension agreement separately to each independently paired index set, since satisfying these requirements for one pairing provides no information about whether another pairing in the same expression is valid.

Additional Requirement of Distinct Symbols

Verification within this area also confirms that each repeated symbol used to denote a distinct contraction is not reused for a different pairing, since reusing a symbol across more than two index positions would create ambiguity about which indices belong to which contraction.

Relationship to Tensor Operation Notation

The multiple contraction area corresponds to expressions in tensor operation notation containing more than one distinct symbol, each repeated exactly once as an upper and once as a lower index, with the number of such distinct repeated symbols directly indicating the number of independent contractions present in the expression.