10.11.5 Tensor Higher Order Component Multisum Pattern

Tensor Higher Order Component Multisum Pattern is the extension of the single summation pattern found in rank-one transformation rules to the case of a higher-rank tensor, in which several independent repeated indices appear simultaneously within the same term, each pairing a matrix factor to a distinct index of the tensor, so that the full transformation formula expands into a nested sum carried out once for every index rather than a single sum carried out once. It captures the way multiple contractions coexist within one expression without interfering with one another, each contributing its own layer of implicit summation dictated by the Einstein summation convention.

Anatomy of the Pattern

Several Independent Repeated Letters

For a tensor with several indices, the multisum pattern introduces one distinct repeated letter for every index, each letter appearing once as an upper index on one factor and once as a lower index on another, with no letter shared between the contractions belonging to different indices.

This tensor with two upper and two lower indices displays a multisum pattern with four distinct repeated letters, each linking one matrix factor to one index of the original tensor.

Nested Rather Than Sequential Sums

The multiple sums implied by the repeated letters are carried out as a nested structure, meaning that for every value of one summed index, every value of the remaining summed indices is also considered, producing a total number of terms equal to the dimension raised to the power of the number of summed indices.

Governing Rules

No Overlap Between Distinct Contractions

Each repeated letter in the multisum pattern must be distinct from every other repeated letter within the same term, since reusing a letter for two different contractions would make it impossible to tell which matrix factor pairs with which index of the original tensor.

Range of Each Sum Set by Dimension

Every individual sum within the multisum pattern ranges independently over every integer from one up to the dimension of the underlying vector space, exactly as a single summation pattern would, with the total expansion obtained by combining all these independent ranges together.

Free Indices Remain Outside the Pattern

Indices that appear only once, without a matching occurrence elsewhere in the term, remain free and are excluded from the multisum pattern entirely, continuing to label which particular component of the new tensor the equation describes.

Consequences of the Pattern

Rapid Growth of Explicit Term Count

Because the multisum pattern involves several independent sums rather than one, the number of explicit terms produced when the pattern is expanded grows multiplicatively with the number of indices, in contrast to the single sum of a rank-one transformation, which grows only linearly with the dimension.

Compact Notation Despite Growing Complexity

Despite this rapid growth in the number of explicit terms, the multisum pattern allows the transformation rule for a tensor of any rank to be written just as compactly as the rank-one rule, since the repeated-index notation absorbs the increasing complexity without requiring additional summation symbols.

Schematic Representation

The diagram highlights two of the several independent summation letters that together make up the multisum pattern, each representing its own implicit sum layered within the same overall expression.