13.15.2 Tensor Contraction Two Index Removal

Tensor Contraction Two Index Removal is the elementary fact that every single application of tensor contraction eliminates exactly two indices from the object being acted upon, one contravariant and one covariant, regardless of the tensor's overall rank or how many other indices remain untouched. It identifies the fixed, invariant quantum of change associated with a single contraction, serving as the basic unit from which the broader order reduction behavior of repeated or multiple contractions is built up.

Conceptual Basis

A Fixed Quantum of Reduction

Unlike operations whose effect on rank might vary depending on circumstances, a single contraction always removes precisely two indices, no more and no fewer, making two index removal a constant and predictable feature of the operation rather than something contingent on the specific tensor involved.

Why Exactly Two and Not One or More

Because a valid contraction requires pairing one contravariant index with one covariant index and summing over their shared range, the operation necessarily consumes both members of the pair simultaneously; it is not possible for a single contraction to remove only one index or to remove more than two, since contraction is defined precisely in terms of this one-to-one pairing.

Independence From the Remaining Indices

Two index removal describes only the change contributed by the pair being contracted, leaving any other indices present on the tensor entirely unaffected, whether those other indices are additional contravariant slots, additional covariant slots, or a mixture of both.

Formal Description

Rank Change From a Single Contraction

For a tensor of type $$(p,q)$$, applying one contraction produces a tensor of type:

with the total order decreasing from $$p+q$$ to $$p+q-2$$, a reduction of exactly two regardless of the specific values of $$p$$ and $$q$$.

Illustration on a Specific Example

Applying a single contraction to a tensor $$T_{kl}^{ij}$$ of type $$(2,2)$$ over the pair $$i,k$$ gives:

producing a tensor of type $$(1,1)$$, confirming that exactly two indices, one of each variance, have been removed.

Repetition as the Basis for Larger Reductions

When several contractions are performed in sequence, the total number of indices removed equals two multiplied by the number of contractions performed, since each individual contraction contributes its own fixed removal of two indices independently of the others.

Properties

Universality Across All Contraction Cases

Two index removal applies without exception to every named contraction case, including trace contraction, matrix multiplication, inner products, and vector covector pairing, since all of these are built from the same underlying single-contraction operation governed by this fixed removal rule.

Consistency With the General Result Type Formula

The general formula describing a tensor's type after $$k$$ contractions is a direct consequence of two index removal applied repeatedly, since each of the $$k$$ contractions independently contributes its own reduction of one contravariant and one covariant index.

Impossibility of Partial or Fractional Removal

Because contraction is defined as pairing and summing over a complete matched pair of indices, it is not possible for a contraction to remove only a fraction of an index or to affect an index without fully consuming both members of its pair, reinforcing that two index removal is an all-or-nothing feature of the operation.

Practical Significance

Building Block for Predicting Rank Reduction

Two index removal serves as the elementary building block used to predict the overall rank reduction resulting from any number of contractions, since the total effect of a sequence of contractions is obtained simply by multiplying this fixed per-contraction removal by the number of contractions applied.

Bookkeeping in Complex Tensor Expressions

In tracking the structure of a tensor expression through several stages of simplification, relying on the fixed two index removal per contraction provides a simple and reliable bookkeeping method for confirming that the expected number of indices remains at each stage.

Distinguishing Contraction From Other Rank-Altering Operations

Recognizing two index removal as a defining signature of contraction helps distinguish it from other tensor operations that may alter rank differently, such as taking an outer product, which increases rather than decreases the total index count, reinforcing the specific and consistent behavior unique to contraction.