14.18.3 Tensor Map Product Basis Coefficient Rule

The Tensor Map Product Basis Coefficient Rule describes how tensor products interact with basis elements through coefficient multiplication in algebraic structures.

Tensor Map Product Basis Coefficient Rule is the rule stating that the scalar coefficient attached to the output basis tensor $e_{a}' \otimes f_{b}'$ in the expansion of $(f \otimes g) (e_{i} \otimes f_{j})$ is obtained by simply multiplying the corresponding coefficient of $f$ by the corresponding coefficient of $g$ , rather than by any more elaborate combination of the two.

Statement of the Rule

The Two Source Coefficients

Fix the expansions of the images of a single basis vector of $V$ under $f$ and a single basis vector of $W$ under $g$ ,

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

so that $F_{i}^{a}$ is the coefficient of $e_{a}'$ in $f (e_{i})$ , and $G_{j}^{b}$ is the coefficient of $f_{b}'$ in $g (f_{j})$ .

The Coefficient Rule Itself

The rule asserts that the coefficient of $e_{a}' \otimes f_{b}'$ in $(f \otimes g) (e_{i} \otimes f_{j})$ is the plain product

C_{i j}^{a b} = F_{i}^{a} \cdot G_{j}^{b}

with no additional terms, no sum of cross products, and no dependence on any coefficient of $f$ or $g$ other than the single pair $F_{i}^{a}$ and $G_{j}^{b}$ matched to the specific indices $i, j, a, b$ in question.

Derivation of the Rule

Substitution Into the Bilinear Tensor Symbol

Applying the tensor product map to $e_{i} \otimes f_{j}$ gives $f (e_{i}) \otimes g (f_{j})$ , and substituting the two expansions turns this into

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

Bilinearity Distributes the Sums

Because the tensor symbol $\otimes$ is bilinear, it distributes over both sums independently and pulls out both scalar coefficients, yielding

\sum_{a} \sum_{b} F_{i}^{a} G_{j}^{b} e_{a}' \otimes f_{b}'

and reading off the coefficient of the single basis tensor $e_{a}' \otimes f_{b}'$ from this sum, for one fixed pair $(a, b)$ , produces exactly the product $F_{i}^{a} G_{j}^{b}$ with no other term contributing, since each pair $(a, b)$ in the double sum labels a distinct basis tensor of the codomain and there is no collapsing between different pairs.

Properties of the Coefficient Rule

Multiplicativity, Not Additivity

Unlike the coefficient rule for a direct sum of maps, where a coefficient of a combined map is a sum of the two separate coefficients, the coefficient rule for a tensor product of maps is multiplicative: the combined coefficient $C_{i j}^{a b}$ vanishes as soon as either factor $F_{i}^{a}$ or $G_{j}^{b}$ vanishes, a behavior that has no analogue in additive combination rules.

Separability of Indices

The rule treats the pair of indices $(i, a)$ , associated with $f$ , entirely independently of the pair $(j, b)$ , associated with $g$ : changing $i$ or $a$ affects only the factor $F_{i}^{a}$ , and changing $j$ or $b$ affects only the factor $G_{j}^{b}$ . This separability is what makes the combined coefficient array the Kronecker product of the two individual matrices rather than some more entangled combination.

Linearity in Each Factor Separately

For fixed $j$ and $b$ , the coefficient $C_{i j}^{a b}$ is a linear function of the entries of $F$ , and symmetrically, for fixed $i$ and $a$ , it is a linear function of the entries of $G$ ; the rule is a single scalar multiplication in each factor separately, which is the coefficient-level expression of the fact that $(f, g) \mapsto f \otimes g$ is itself bilinear in the pair of maps.

Consequence for the Combined Matrix

Entrywise Kronecker Product

Collecting the coefficients $C_{i j}^{a b}$ into a matrix indexed by the compound row label $(a, b)$ and compound column label $(i, j)$ reproduces exactly the Kronecker product $F \otimes G$ entry by entry, so that the coefficient rule is the elementary statement underlying the block structure of the Kronecker product: each block of the Kronecker product is a copy of $G$ scaled by a single entry of $F$ .