8.20.1 Tensor Implicit Sum Expansion

Tensor Implicit Sum Expansion is a notation technique in tensor algebra that simplifies the representation of sums by implicitly indicating repeated indices.

Tensor Implicit Sum Expansion is the specific sub-step of the index expansion operation that targets only the summation implied by the Einstein convention, converting a term in which an index is repeated once as a superscript and once as a subscript into the explicit sum-of-products form, without necessarily also expanding any free indices present in the same expression. It isolates the "implicit" part of index notation — the summation sign that convention allows to be omitted — and makes that omitted sign and every one of its summands visible.

What Counts as Implicit

The Omitted Summation Sign

Under the Einstein summation convention, a term such as A^i B_i is understood to mean the same as Σᵢ A^i B_i, but the summation symbol Σ and its range are not written; they are implicit, inferred purely from the fact that the letter i occurs twice with opposite variance in the same term. Implicit sum expansion is the operation of restoring the summation sign and the explicit list of terms that the convention allows to be suppressed:

A^{i} B_{i} \equiv \sum_{i = 1}^{n} A^{i} B_{i} = A^{1} B_{1} + \dots + A^{n} B_{n}

Distinguishing Implicit From Explicit Sums

Not every sum appearing in a tensor expression is implicit in this sense: an explicitly written summation sign, or a sum over an index that does not follow the repeated upper-lower pattern, is already explicit and requires no expansion of this particular kind. Implicit sum expansion applies strictly to sums that are present only because of the repeated-index convention and would otherwise be invisible in the written expression.

Identifying Which Indices Qualify

The Repetition-and-Variance Test

Before expansion can be performed, each index in a term must be checked against two conditions: it must occur exactly twice in that term, and the two occurrences must be one upper and one lower. An index occurring three or more times, or occurring twice with the same variance, does not qualify for implicit summation under the standard convention and signals either a notational error or a context using an extended, non-standard convention.

Term-by-Term Scope

The implicit summation convention, and hence implicit sum expansion, applies separately within each additive term of a larger expression. In A^i B_i + C^j D_j E^k F_k, the index i is summed within the first term, and j and k are each summed independently within the second term; expansion is carried out term by term, and indices repeated across different additive terms (rather than within the same term) are not summed at all.

Nested Implicit Sums

Several Repeated Indices in One Term

A single term can contain more than one repeated upper-lower pair, in which case implicit sum expansion introduces one nested sum for each pair, and the fully expanded form contains a number of scalar product terms equal to the product of the summed dimension for each pair:

A^{i} B_{i}^{j} C_{j} = \sum_{i = 1}^{n} \sum_{j = 1}^{n} A^{i} B_{i}^{j} C_{j}

expanding into n² individual scalar products before they are added, one for each combination of values of i and j.

Order of Expansion Does Not Affect the Result

Because addition and multiplication of scalars are commutative and associative, the nested sums produced by expanding several implicit summations in the same term may be carried out in any order, and grouped or regrouped freely, without changing the final total. This is what allows nested implicit sums to be treated, after expansion, as ordinary finite double or multiple sums subject to the usual rules for reordering summation.

Diagram of a Nested Implicit Sum Expanding Outward

Practical Role in Verifying Contractions

Confirming a Contraction Was Applied Correctly

Because implicit sum expansion makes every summand visible, it is the standard tool for checking, in a concrete low-dimensional example, that a contraction claimed in index notation has been performed correctly — that the right index was summed, over the right range, against the right partner index. Expanding suspect terms explicitly and comparing against an independently computed matrix or vector result is a direct way to catch an index placed with the wrong variance or a summation mistakenly omitted or duplicated.

Relationship to General Index Expansion

Implicit sum expansion is one of the two component operations that together make up the full index expansion operation described more generally; the other component operation, expansion of free indices into separate equations, is applied independently and can precede or follow implicit sum expansion within the same overall expansion of a compound tensor expression.