13.8.2 Tensor Trace Contraction Diagonal Summation

Tensor Trace Contraction Diagonal Summation is the explicit componentwise procedure of adding together the entries of a mixed tensor for which the contravariant and covariant index values coincide, providing the concrete computational realization of the abstract trace contraction operation once a specific basis and specific component values have been fixed. It describes the mechanical act of summation performed over the diagonal entries, as distinct from the coordinate-free statement that the trace is obtained by contracting a mixed index pair.

Conceptual Basis

From Abstract Contraction to Concrete Sum

The trace contraction case is defined abstractly as summing over a repeated mixed index using the Einstein convention. Diagonal summation is what this abstract statement becomes once the tensor's components are written out explicitly in a chosen basis, reducing the operation to an ordinary sum of numbers.

Origin of the Term Diagonal

For a rank-two mixed tensor displayed as a square array of components, the entries for which the row index equals the column index lie along the main diagonal of that array. Diagonal summation names the operation of adding exactly these entries together.

Dependence on a Fixed Index Range

Diagonal summation requires that the shared index range over a finite set of values corresponding to the dimension of the underlying vector space, since the sum is carried out explicitly over each value that index can take.

Formal Description

The Summation Formula

For a mixed tensor $$T_j^i$$ defined over an $$n$$-dimensional space, diagonal summation is written explicitly as:

listing each term $$T_1^1,T_2^2,\dots ,T_n^n$$ individually before their sum is taken.

Relation to the Einstein Convention

The Einstein summation convention abbreviates the explicit sum above by omitting the summation symbol whenever an index appears once as a superscript and once as a subscript in the same term, so that $$T_i^i$$ alone is understood to denote the full diagonal summation.

Numerical Example

For a two-dimensional space with components $$T_1^1=3$$ and $$T_2^2=5$$, the diagonal summation yields:

regardless of the values taken by the off-diagonal components.

Properties

Independence From Off-Diagonal Entries

Diagonal summation depends only on the components for which the two indices coincide, meaning off-diagonal entries of the tensor, however large or structured, contribute nothing directly to the resulting sum.

Consistency With Basis Independence

Although diagonal summation is carried out using the specific components of a chosen basis, the resulting numerical value agrees with the sum obtained in any other basis, since the trace contraction underlying this computation is basis-independent.

Computational Simplicity

Diagonal summation requires only as many additions as the dimension of the underlying space, making it computationally straightforward relative to operations that involve the full array of off-diagonal components.

Extensions

Partial Diagonal Summation in Higher-Rank Tensors

When a higher-rank tensor has one designated mixed pair selected for tracing while other indices remain free, diagonal summation is applied only across the values of the selected pair, with the remaining free indices held fixed for each term of the sum.

Diagonal Summation Across Multiple Pairs

If several mixed pairs are contracted simultaneously, diagonal summation generalizes to a nested sum, running independently over each contracted pair's shared index while combining the corresponding components into a single accumulated total.

Role in Numerical Implementation

In computational settings where tensors are represented as explicit arrays, diagonal summation corresponds directly to iterating over the array's diagonal positions and accumulating their values, making it the practical algorithmic counterpart of the trace contraction operation.