14.17.4 Tensor Map Product Component Summation Case

The Tensor Map Product Component Summation Case explains how tensor components combine under product mappings through summation over shared indices.

Tensor Map Product Component Summation Case is the situation in which the component formula for a tensor product map $f \otimes g$ must be applied not to a single simple tensor $v \otimes w$ but to a general element of $V \otimes W$ that is itself a sum of simple tensors, forcing the evaluation of $(f \otimes g)$ to pass through an explicit double summation over basis indices rather than a single product of components.

Setup

General Elements Are Sums of Simple Tensors

A generic element $t$ of $V \otimes W$ , expressed with respect to bases ${e_{i}}$ of $V$ and ${f_{j}}$ of $W$ , is not in general a single simple tensor $v \otimes w$ , but a sum

t = \sum_{i} \sum_{j} t^{i j} e_{i} \otimes f_{j}

where the coefficients $t^{i j}$ form a two-index array that need not factor as a product of a vector of $i$ -indices and a vector of $j$ -indices. This is precisely the situation that requires the summation case of the component formula.

The Component Formula on a Single Basis Tensor

On a single basis tensor $e_{i} \otimes f_{j}$ , the tensor product map still acts by the simple product rule

(f \otimes g) (e_{i} \otimes f_{j}) = \sum_{a} \sum_{b} F_{i}^{a} G_{j}^{b} e_{a}' \otimes f_{b}'

with the sums running once over the target index $a$ and once over the target index $b$ . The summation case arises when this rule must additionally be summed over the source indices $i$ and $j$ because the input itself is a sum.

The Full Summation Formula

Applying the Map to a General Tensor

Substituting the expansion of $t$ and using linearity of $f \otimes g$ gives

(f \otimes g) (t) = \sum_{i} \sum_{j} \sum_{a} \sum_{b} t^{i j} F_{i}^{a} G_{j}^{b} e_{a}' \otimes f_{b}'

so that the resulting coefficient attached to $e_{a}' \otimes f_{b}'$ in the image is

s^{a b} = \sum_{i} \sum_{j} F_{i}^{a} G_{j}^{b} t^{i j}

a genuine double sum over both source indices, in contrast with the single-tensor case where each output coefficient was simply a product of one component of $F$ and one component of $G$ .

Einstein Summation Convention Form

Under the Einstein summation convention, in which repeated upper and lower indices are automatically summed, the same formula is written compactly as

s^{a b} = F_{i}^{a} G_{j}^{b} t^{i j}

with the indices $i$ and $j$ understood as summed, since each appears once as a subscript and once as a superscript across the three factors.

Matrix Form of the Summation Case

The Array as a Matrix

When $t^{i j}$ is arranged as a matrix $T$ with row index $i$ and column index $j$ , the double-summation formula for $s^{a b}$ is exactly the matrix product

S = F^{T} T G

where $F^{T}$ denotes the transpose of the component matrix of $f$ , so that the summation case reduces the action of $f \otimes g$ on a general tensor, represented as a matrix rather than a flattened vector, to two ordinary matrix multiplications instead of a single Kronecker product acting on a long column vector.

Equivalence With the Kronecker Product Formula

Flattening $T$ into a single column vector by stacking its columns reproduces the same result obtained by multiplying that column vector by the Kronecker product $G \otimes F$ , confirming that the summation-case formula and the single Kronecker product formula compute the same linear map, differing only in whether the general tensor is treated as a two-index matrix or as a single flattened vector of coefficients.

Special Cases Within the Summation Case

Reduction to the Simple Tensor Case

If the array $t^{i j}$ factors as $t^{i j} = v^{i} w^{j}$ for some vectors $v$ and $w$ , the double sum collapses back into the single-tensor product rule,

s^{a b} = (\sum_{i} F_{i}^{a} v^{i}) (\sum_{j} G_{j}^{b} w^{j})

confirming that the simple-tensor case treated elsewhere in the tensor map product component formula is the special case of the summation case in which the coefficient array has rank one.

Rank as a Measure of Genuine Summation

More generally, the number of simple tensors needed to write $t$ as a sum, which coincides with the rank of the matrix $T$ , measures how far the situation is from the simple product case: rank one recovers a single product of components, while higher rank strictly requires the full double-summation formula to compute the image under $f \otimes g$ .