14.15.1 Tensor Map Product Composable Factor Maps

Tensor Map Product Composable Factor Maps explore how tensor maps compose through product structures, enabling algebraic transformations in multilinear contexts.

Tensor Map Product Composable Factor Maps is the requirement that, within each individual factor slot of a combined operator composition, the codomain of the earlier map must match the domain of the later map, since the tensor product composition rule only applies once every factor-level composition is itself well defined.

The Composability Condition

Matching Domain and Codomain Within Each Factor

For a composition of two combined operators to be defined through the composition rule, the map applied second in the first factor must have a domain equal to the codomain of the map applied first in that same factor, and likewise for the second factor.

S_{1} : W_{1} \to U_{1}

T_{1} : V_{1} \to W_{1}

Composability Required in Every Factor Independently

This composability condition must hold separately for each factor slot; a mismatch in only one factor's domain and codomain is enough to make the overall composition of combined operators undefined, even if the other factor's maps compose without issue.

Consequence for the Combined Operators

Well-Defined Composition of Combined Operators

Only when both factors satisfy the composability condition individually does the composition of the two combined operators make sense, since the combined operator built from S₁ and S₂ requires its domain, the tensor product of the codomains of T₁ and T₂, to match exactly the tensor product of the domains of S₁ and S₂.

(S_{1} \otimes S_{2}) \circ (T_{1} \otimes T_{2}) : V_{1} \otimes V_{2} \to U_{1} \otimes U_{2}

Diagram of the Composability Requirement

Matching Codomain to Domain in Each Factor

The diagram below shows the codomain of the first map in each factor matching exactly the domain of the second map in that same factor, a condition required independently in both factor slots.

Consequences of a Composability Failure

Undefined Composition in a Single Factor Blocks the Whole Construction

If the codomain of T₁ fails to match the domain of S₁, the factor-level composition S₁ composed with T₁ is not defined, and consequently the composition of the two combined operators built from these factor maps is not defined either, regardless of whether the second factor's maps compose correctly.

No Partial Composition Across Mismatched Factors

There is no meaningful partial version of the composed combined operator when only one factor is composable; the tensor product construction requires both factor-level compositions to be simultaneously well defined before the combined operator's composition can be formed at all.

Verifying Composability in Practice

Checking Spaces Before Attempting Composition

Before applying the composition rule, it is necessary to verify, for each factor separately, that the relevant domain and codomain spaces coincide, typically by comparing the specific vector spaces involved or, once bases are fixed, by comparing the corresponding matrix dimensions.

Dimension Matching as a Necessary but Not Sufficient Check

Matching dimensions between the codomain of one factor map and the domain of the next is necessary for composability but not sufficient on its own, since two spaces of the same dimension may still be regarded as distinct spaces requiring an explicit identification before composition is meaningful.

Matrix-Level Composability

Conformable Factor Matrices

At the matrix level, composability of factor maps corresponds to conformability of the corresponding factor matrices for ordinary matrix multiplication, meaning the number of columns of the earlier matrix in each factor must equal the number of rows of the later matrix in that same factor.

cols (T_{1}) = rows (S_{1})

Composite Matrix Product Requires Both Factors Conformable

The Kronecker product formulation of the composition rule for matrices is valid only once conformability is confirmed separately for each factor's pair of matrices, mirroring the abstract composability condition on the underlying linear maps.

Extension to Several Factors

Composability Required in Every Slot Simultaneously

When the tensor product involves three or more factor spaces, composability must be verified independently in every individual factor slot, since a single mismatched pair of maps in any one factor is enough to leave the entire composed combined operator undefined.

Building Composability Incrementally

In practice, composability across many factors is often verified incrementally, checking one factor at a time, since confirming composability in a subset of the factors does not by itself guarantee composability in the remaining factors, which must be checked separately.