13.4 Tensor Single Index Contraction Operation

Tensor Single Index Contraction Operation is the specific instance of the tensor contraction operation in which exactly one contravariant index and one covariant index are selected and summed, treated as a distinct, elementary operation from which more elaborate combinations of contractions are built.

Defining Features of the Operation

Exactly One Pair Involved

The single index contraction operation is characterized by acting on precisely one contravariant slot and one covariant slot, with every other index present on the input tensor left untouched by this particular application of the operation.

Result of a Fixed Type Reduction

A single index contraction operation always reduces the input tensor's type by exactly one contravariant index and one covariant index, producing an output of order two less than the input, regardless of how many indices the input originally carried in total.

Steps Composing This Operation

Slot Pair Selection

The operation begins by selecting one contravariant and one covariant slot from among those available on the input tensor, a step that must precede any summation since the pair to be summed must first be identified.

Index Summation

Following selection, the operation proceeds by summing the tensor's components over every value the selected shared index can take, producing the numerical entries of the resulting lower-order tensor.

The Operation as a Building Block

Serving as the Unit of Repeated Application

Because larger combinations of contractions, such as multiple independent pairings or full contraction to a scalar, can be understood as several applications of the single index contraction operation performed on successive or simultaneous pairs, this operation functions as the elementary unit from which those more elaborate cases are constructed.

Composability Across Successive Applications

The output of one single index contraction operation, provided it still retains at least one contravariant and one covariant index, can itself serve as the input to a subsequent application of the same operation, allowing repeated reduction toward a scalar through a sequence of elementary steps.

Requirements Specific to This Operation

Applicability Condition on the Input

The single index contraction operation can be applied to any tensor possessing at least one contravariant and one covariant index, without further restriction on the tensor's total order, provided a valid pair satisfying opposite variance and matching dimension can be identified among its slots.

Independence from Other Indices Present

Because the operation is confined to exactly one pair, its validity and outcome do not depend on the number or configuration of any additional free indices the input tensor may carry, which remain entirely unaffected by this particular application.

Relationship to Tensor Operation Notation

The single index contraction operation is denoted by exactly one index symbol appearing as both an upper and lower index within a term, distinguishing it in notation from expressions containing more than one such repeated symbol, which instead denote multiple applications of this elementary operation carried out together.