13.15 Tensor Contraction Order Reduction Behavior

Tensor Contraction Order Reduction Behavior is the systematic pattern by which the rank, or order, of a tensor decreases as contraction operations are applied to it. It characterizes how the total number of indices, and the split between contravariant and covariant indices, changes predictably each time a contraction is performed, regardless of the specific tensor components involved.

Definition

For a tensor $$T$$ of order $$r=p+q$$, where $$p$$ counts contravariant indices and $$q$$ counts covariant indices, a single contraction operation acting on one upper and one lower index produces a tensor of order:

This order reduction behavior is deterministic: it depends solely on the number of contractions applied, not on the tensor's numerical content or the underlying vector space's dimension.

Single Contraction Behavior

Mechanism

A contraction sets one upper index equal to one lower index and sums over the shared range according to the Einstein summation convention.

The repeated index $$a$$, appearing once up and once down, is summed over all its allowed values, and both occurrences vanish from the free-index list of the result.

Immediate Effect

Each contraction removes exactly two free indices: one contravariant and one covariant. The order therefore decreases by two per contraction, never by one, and never by more than two in a single step.

Iterated Contraction Behavior

Sequential Order Drop

Applying $$k$$ independent contractions to a tensor of order $$r$$ produces a residual tensor of order:

provided that $$2k\le r$$ and that enough upper and lower indices remain available at each step to perform the next contraction.

Terminal Cases

If the tensor originally has an equal number of upper and lower indices, $$p=q$$, then full iterated contraction can drive the order down to zero, producing a scalar. If $$p\ne q$$, contraction halts once either the upper or lower index supply is exhausted, leaving a residual tensor with only one variance type remaining.

Non-Reversibility

Order reduction through contraction is not reversible: no unique tensor of the original order can be recovered from the contracted result, since the summation over the shared index discards information about how the removed components were distributed.

Behavior Across Multiple Independent Contractions

When two disjoint pairs of indices are contracted simultaneously, order reduction is additive: each pair contributes an independent drop of two, and the total order reduction equals four, regardless of the order in which the two contractions are carried out.

This additivity holds because contractions on disjoint index pairs commute: performing them in either order yields the same residual tensor.

Visual Summary

Relation to Rank and Dimension

Order reduction behavior is independent of the dimension $$n$$ of the underlying vector space. Whether $$n=2$$ or $$n=100$$, a single contraction always removes exactly two from the order count, even though the number of terms summed in the contraction depends directly on $$n$$.