13.5.5 Tensor Multiple Contraction Result Structure

Tensor multiple contraction results yield scalars through repeated index contraction, simplifying complex tensor expressions.

Tensor Multiple Contraction Result Structure is the arrangement of free indices, variance, and dimension characterizing the tensor produced by a multiple index contraction operation, describing the shape of the output that remains once every pair in the associated slot pair set has been contracted away.

Composition of the Result Structure

Indices Surviving Outside the Slot Pair Set

The result structure consists of every index of the input tensor that does not belong to any member of the slot pair set, since these are precisely the positions left untouched by the several contractions comprising the operation.

T_{i j}^{i j k} \to S_{k}

Preservation of Variance and Dimension

Each surviving index retains exactly the variance and dimension it possessed on the input tensor, since none of the several contractions act upon slots outside their own designated pairs, leaving every other slot entirely unaffected.

Order of the Result Structure

Reduction by Twice the Number of Pairs

The order of the result structure equals the order of the input tensor reduced by twice the number of pairs in the slot pair set, reflecting the combined effect of each individual pair's contribution of a reduction of two.

order (result) = order (T) - 2 k

Here $k$ denotes the number of pairs comprising the slot pair set applied.

The Endpoint of Zero Remaining Order

When the slot pair set happens to constitute an all slot pairing, exhausting every index of the input tensor, the result structure contains no free indices at all, coinciding with the structure of a scalar produced by the full contraction operation.

Structural Independence from Numerical Computation

Determined Before Any Summation Is Carried Out

The result structure is fully fixed once the input tensor's structure and the chosen slot pair set are known, independent of the specific numerical values that the associated summation set will later produce for each surviving component.

Complementary Relationship to the Summation Set

The result structure and the summation set together account for every index of the input tensor, with the summation set identifying the indices removed through contraction and the result structure identifying the indices that persist into the output.

Result Structure Across Different Slot Pair Sets

Dependence on the Specific Pairing Chosen

Because different slot pair sets may leave different sets of indices free, even when applied to the same input tensor and consisting of the same number of pairs, the specific result structure produced depends on precisely which slot pair set was selected, not merely on the number of pairs it contains.

T_{i k}^{i l} \to S_{k}^{l}

Consistency Given a Fixed Slot Pair Set

Once a particular slot pair set is fixed, the result structure it produces is uniquely determined and does not vary with the order in which the individual pairs within that set are contracted, consistent with the independence of pair ordering established for disjoint contractions.

Relationship to Tensor Operation Notation

The multiple contraction result structure is read directly from tensor operation notation by identifying which index symbols remain unrepeated once every matched upper-lower pair specified by the slot pair set has been accounted for, since these singly appearing symbols, together with their variance, constitute the notated description of the resulting structure.