8.19.3 Tensor Contraction Summed Dimension

Tensor contraction sums dimensions by combining indices, simplifying tensor expressions through index contraction in algebraic operations.

Tensor Contraction Summed Dimension is the range over which a contracted index runs when a tensor contraction is carried out, equal to the dimension n of the underlying vector space on which the tensor is defined, and it determines both the number of terms folded into a single contracted component and the size of the identity that a fully contracted Kronecker delta produces. It is the quantitative counterpart to the qualitative contraction pattern: the pattern says a repeated upper-lower index is summed, and the summed dimension says exactly how many values that index takes as it is summed.

Definition in Terms of the Underlying Space

Dimension as the Range of Every Index

If a tensor is defined on an n-dimensional vector space V, every index of every tensor built from V and its dual V* ranges over 1 through n (or 0 through n-1, depending on convention). Consequently, a contraction such as

T^{i} S_{i} = \sum_{i = 1}^{n} T^{i} S_{i}

expands into exactly n terms, one for each value the contracted index takes, and the summed dimension n is precisely this term count.

Independence From Tensor Rank

The summed dimension depends only on the vector space the indices range over, not on how many indices the tensor has in total. A rank-2 tensor and a rank-10 tensor built on the same n-dimensional space have contractions that each sum over the same n values per contracted index pair; rank affects how many free indices remain after contraction, while the summed dimension affects only how many terms each individual contraction folds together.

Consequence for the Trace

Trace of the Identity

A canonical illustration of the summed dimension is the full contraction of the mixed Kronecker delta with itself, δ^i_i, which does not vanish or reduce to a single value but equals the summed dimension exactly:

δ_{i}^{i} = \sum_{i = 1}^{n} 1 = n

since δ^i_i equals 1 for each of the n values of i and these n ones are added together. This identifies the trace of the identity map as being equal to the summed dimension, matching the elementary linear-algebra fact that the identity matrix's trace equals the dimension of the space it acts on.

Dimension-Dependent Scalar Identities

Any fully contracted expression built purely from the metric and its inverse, such as g^{ij} g_{ij}, likewise evaluates to the summed dimension n, because g^{ij} g_{jk} = δ^i_k reduces the expression to δ^i_i = n. Such identities are used to track dimension-dependent numerical factors that appear when manipulating contracted tensor expressions, particularly in computations that must later be specialized to a fixed number of spatial or spacetime dimensions.

Behavior Under Restriction to Subspaces

Contraction on a Subspace of Lower Dimension

If a tensor is restricted to act on an m-dimensional subspace of an ambient n-dimensional space, with m < n, a contraction carried out entirely within that subspace sums only over the m values available there, so the effective summed dimension for that restricted contraction is m, not the ambient n. This distinction matters whenever a calculation mixes tensors defined on a full space with tensors defined only on an invariant subspace, since the number of terms contributed by each contraction differs accordingly.

Diagram of Terms Folded by Summation

Role in Repeated and Multiple Contractions

Multiple Independent Summed Dimensions

When an expression contracts several distinct index pairs at once, each pair contributes its own sum over the summed dimension, and the total number of scalar terms combined in the fully expanded expression is the product of the summed dimension raised to a power equal to the number of independent contracted pairs. For two independent contractions on an n-dimensional space, an expression such as A^{ij} B_{ij} expands to n² scalar products before they are added together, since both i and j independently range over n values.

Practical Bookkeeping in Computation

Because the summed dimension fixes exactly how many terms a symbolic contraction expands into, it is the quantity used to bound the computational cost of evaluating a contracted expression numerically: a single contraction over an n-dimensional index costs O(n) additions per free-index combination, and nested contractions multiply this cost by the summed dimension for each additional contracted pair, which is the same counting principle that governs the cost of matrix multiplication expressed as an index contraction.