13 Tensor Contractions

Tensor Contractions is the branch of tensor algebra concerned with the operation of summing a tensor's components over one contravariant index and one covariant index set equal to each other, thereby reducing the tensor's order by two and producing a new tensor, or a scalar, that captures a specific relationship encoded within the original tensor.

The Contraction Operation

Basic Definition

Contraction takes a tensor possessing at least one upper index and one lower index, sets those two indices equal to a common summation index, and sums the resulting components over every value that index can take, producing a tensor with two fewer indices than the original.

Requirement of Mixed Variance

Contraction requires that the two indices being summed have opposite variance, one contravariant and one covariant, since this pairing is what allows the summation to remain invariant under a change of basis, a property that would fail if two indices of the same variance were summed together.

Reduction in Order

Removal of Two Indices at a Time

Each application of contraction removes exactly one upper index and one lower index from a tensor, so that a tensor of a given order becomes a tensor of an order reduced by two after a single contraction is performed.

Repeated Contraction Toward a Scalar

A tensor may be contracted repeatedly, on different pairs of indices in turn, and if the tensor's order is even and the process is applied fully, repeated contraction can reduce the tensor all the way down to a scalar, a single invariant value with no remaining free indices.

Contraction Applied to a Product of Tensors

Combining Contraction with the Tensor Product

Contraction is frequently applied to an index from one tensor and an index from another after those tensors have first been combined through the tensor product, producing an operation that behaves similarly to matrix multiplication or the action of a linear map on a vector.

Free Indices Alongside a Contraction

When contraction acts on only some of the indices present in a product of tensors, the remaining indices persist as free indices of the resulting tensor, so that a single expression can contain both a contraction and free indices simultaneously.

Structural Requirements Governing Contraction

Index Slot and Type Requirements

For a contraction to be defined, the two indices selected must genuinely exist as slots on the tensor or tensors involved and must carry the required opposite variance, conditions that are examined during the verification checks applied before a contraction is carried out.

Dimension Agreement Between Contracted Indices

The two indices being contracted must range over spaces of the same dimension, since the summation pairs values of the two indices term by term and requires both to run over an identical range.

Invariance of the Result

Preservation Under Change of Basis

A defining property of contraction is that its result transforms correctly under a change of basis given the transformation behavior of the original tensor, with the case of contracting all indices of a tensor down to a scalar yielding a value that remains entirely unchanged regardless of the basis chosen.

Relationship to Tensor Operation Notation

Tensor contraction is expressed entirely through the repetition of an index symbol as both an upper and a lower index within a single term of an expression, relying on the summation convention established within tensor operation notation to indicate, without an explicit summation symbol, that the repeated index is to be summed over its full range.