8.20.3 Tensor Component Sum Expansion

Tensor Component Sum Expansion simplifies tensor expressions by summing over indices, revealing underlying algebraic structures in multilinear algebra.

Tensor Component Sum Expansion is the sub-step of the index expansion operation that writes out, for one fixed assignment of values to the free indices of a tensor expression, the explicit numerical or symbolic sum of the individual components that combine to produce that single component of the result. It focuses on the destination side of an index equation — a particular component of the output tensor — and shows exactly which components of the input tensors, multiplied together, are added to produce that one output number.

Isolating a Single Output Component

Fixing the Free Indices First

Given an expression such as v^i = A^i_j w^j, component sum expansion begins by fixing the free index i to one particular value, say i = 2, and then expands the implicit sum over the dummy index j for that fixed value alone:

v^{2} = A_{1}^{2} w^{1} + A_{2}^{2} w^{2} + A_{3}^{2} w^{3}

This single equation is the component sum expansion for the output component v^2; it says nothing about v^1 or v^3, each of which has its own, separately written, component sum expansion.

Relation to Full Index Expansion

Full index expansion of an expression with free indices consists of exactly one component sum expansion for every combination of values the free indices can take; component sum expansion is therefore the atomic unit from which the complete expanded set of equations is assembled, one output component at a time.

Structure of a Single Component's Sum

Product Terms Contributing to One Component

Each term inside a component sum expansion is a plain numerical or symbolic product of specific tensor components, one component drawn from each tensor factor in the original expression, with the dummy index replaced by one particular value in every term. The number of such product terms in a single component's expansion equals the summed dimension raised to the power of however many independent dummy indices are being summed in that term.

Multiple Contracted Indices Within One Component

When the original expression contracts more than one pair of indices while leaving some indices free, the component sum expansion for a single fixed combination of free-index values still requires expanding every dummy index nested within it. For T^i = A^i_{\ jk} B^j C^k, fixing i to a specific value produces a component sum expansion with one term for every combination of values of j and k:

T^{1} = \sum_{j = 1}^{n} \sum_{k = 1}^{n} A_{j k}^{1} B^{j} C^{k}

with n² individual product terms folded into the single scalar T¹.

Use in Manual and Numerical Verification

Checking One Component at a Time

Component sum expansion is the natural granularity for manually verifying a computed or claimed tensor result: rather than expanding an entire tensor equation for all combinations of free indices simultaneously, a single suspect component can be checked in isolation by expanding only its own sum, which limits the arithmetic to be verified to exactly the terms that contribute to that one number.

Correspondence to a Single Loop Iteration in Computation

In a numerical implementation of a contraction, computing one component of the output tensor corresponds exactly to performing one component sum expansion: a loop over the free indices selects which output component is being computed, and, for the value of the free indices fixed by that loop iteration, an inner loop (or nested loops, for multiple dummy indices) accumulates the terms of the component sum expansion into that one output entry.

Diagram of One Component's Expansion

Distinguishing Component Sum Expansion From Related Operations

Contrast With Free-Index Expansion

Free-index expansion produces the entire list of separate equations, one per output component; component sum expansion refers to any single one of those equations, taken on its own, together with the full expansion of whatever dummy indices appear inside it. The two are complementary views of the same underlying full expansion: free-index expansion asks "how many equations are there," while component sum expansion asks "what does this one equation say."

Contrast With Implicit Sum Expansion

Implicit sum expansion targets the general operation of writing out a repeated-index summation without necessarily fixing which output component is under consideration; component sum expansion applies that same operation but specifically after the free indices have already been fixed to concrete values, so that the result is a sum of plain numbers or symbolic constants rather than a sum still containing an unresolved free index.