13.3.3 Tensor Contraction Index Summation

Tensor Contraction Index Summation simplifies tensors by summing over repeated indices, key to algebraic manipulation and physical applications.

Tensor Contraction Index Summation is the specific arithmetic step within the tensor contraction operation in which the components of a tensor, indexed by a shared contravariant and covariant index, are added together across every value that shared index can take, yielding the numerical values of the contracted result.

The Mechanics of the Summation

Ranging Over the Full Dimension

Index summation proceeds by letting the shared index take every integer value from one up to the dimension of the vector space it ranges over, adding the corresponding component of the tensor at each successive value.

T_{i}^{i} = \sum_{i = 1}^{n} T_{i}^{i}

The Summation Convention as Implicit Instruction

Under the summation convention, the explicit summation symbol is omitted, and the repetition of an index symbol as both an upper and lower position within a single term is itself the instruction to carry out this summation over the full range of that index.

T_{i}^{i} \equiv \sum_{i = 1}^{n} T_{i}^{i}

Summation in the Presence of Free Indices

Repeating the Summation for Each Free Index Combination

When free indices remain alongside the contracted pair, index summation is carried out separately for every combination of values the free indices may take, producing one summed value for each such combination and thereby populating every component of the resulting lower-order tensor.

A_{i}^{i} j^{} = \sum_{i = 1}^{n} A_{i}^{i j}

Free Indices Held Fixed During Each Summation

While the summation over the contracted index proceeds, every free index is held fixed at a particular value, so that the summation acts only on the designated pair of positions and leaves the values of the free indices untouched during that particular pass of the summation.

Summation Across Multiple Independent Contractions

Nested Application of Separate Summations

When more than one independent contraction is present in an expression, index summation is applied separately for each pair, with the summation over one pair nested within, or performed alongside, the summation over another, since the two summations act on distinct index symbols and do not interfere with one another.

\sum_{i = 1}^{n} \sum_{j = 1}^{n} T_{i j}^{i j}

Commutativity of Independent Summations

Because independent summations act on separate index symbols, the order in which the nested summations are carried out does not affect the final result, allowing the sums over different contracted pairs to be evaluated in either order or combined into a single joint summation.

Summation as the Locus of Component Verification

Where Component-Level Checks Apply

Because index summation is the step at which actual numerical values are combined, any value-dependent requirement examined during component verification, such as a required symmetry among the tensor's entries, is tested against the values entering or produced by this summation step specifically.

Distinguishing the Summation from the Surrounding Operation

Index summation refers narrowly to the arithmetic accumulation of terms, while the broader contraction operation additionally includes the prior selection and pairing of the indices being summed, so that index summation represents the final computational stage of the operation once the pairing has already been established.

Relationship to Tensor Operation Notation

Index summation is denoted in tensor operation notation either through an explicit summation symbol with a stated range or, more commonly, through the repetition of a single index symbol as a matched upper and lower pair, with the summation convention establishing that this repeated notation alone fully specifies the summation to be carried out.