13.3 Tensor Contraction Operation

Tensor Contraction Operation is the specific procedure by which one contravariant index and one covariant index of a tensor are identified, set equal to a shared summation index, and summed over the full range of that index, taken as a single well-defined algebraic operation applied to a tensor or product of tensors.

Steps Constituting the Operation

Selection of the Index Pair

The operation begins with the selection of exactly one upper index slot and one lower index slot from among the indices available on the tensor or tensors involved, a choice that must be made explicit before any summation can proceed.

Assignment of a Shared Index Symbol

Once selected, the two slots are assigned a common index symbol, signaling under the summation convention that these two positions are to be treated as linked for the purpose of the operation.

Summation Over the Shared Index

The operation concludes with summing the components of the tensor over every value the shared index can take, producing a single combined result from what were previously separate indexed components.

Input and Output of the Operation

The Tensor Accepted as Input

The operation accepts as input any tensor possessing at least one contravariant index and at least one covariant index, since the operation cannot be defined on a tensor lacking one of the two required index types.

The Tensor Produced as Output

The operation produces as output a tensor whose order is exactly two less than that of the input, retaining every index of the input other than the selected pair, with those remaining indices preserving their original variance and dimension.

Requirements Governing the Operation

Opposite Variance of the Selected Pair

The operation requires that the two selected indices carry opposite variance, one contravariant and one covariant, since this pairing is what allows the summation to remain consistent with the transformation behavior expected of a tensor.

Equal Dimension of the Selected Pair

The operation requires that the two selected indices range over vector spaces of the same dimension, since the summation defining the operation pairs values of the two indices term by term and therefore requires both to run over an identical set of values.

The Operation as a Linear Map on Tensors

Linearity in the Input Tensor

Viewed as a function from tensors to tensors, the contraction operation is linear in its input, since scaling the input tensor by a constant or adding two input tensors together produces an output related to the individual outputs by the same scaling or addition.

Basis Independence of the Underlying Operation

While the numerical values manipulated during the operation depend on the basis chosen to express the input tensor's components, the operation itself, as an algebraic procedure, is defined without reference to any particular basis, with the output guaranteed to represent the same underlying tensor regardless of the basis used to carry out the computation.

Relationship to Tensor Operation Notation

The contraction operation is expressed entirely through the repetition of a single index symbol as both an upper and lower index within a term of tensor operation notation, with the summation convention supplying the understanding that this repetition denotes the complete procedure of selection, linking, and summation constituting the operation.