14.15 Tensor Map Product Composition Rule
The Tensor Map Product Composition Rule defines how tensor maps combine through multiplication, establishing a framework for operations in multilinear algebra.
Tensor Map Product Composition Rule is the general rule stating that composing two tensor products of maps, each built from one map per factor, equals the tensor product of the compositions of the corresponding factor maps, reducing composition of large combined operators to composition of their smaller individual pieces.
Statement of the Rule
General Formula
Given two combined operators built from pairs of maps on the same two factor spaces, composing the combined operators equals combining the compositions of the corresponding factor maps.
Requirement for the Rule to Apply
The rule requires that the codomain of each map T₁ and T₂ match the domain of the corresponding map S₁ and S₂, so that each individual factor-level composition is well defined before the tensor product is formed.
Verification of the Rule
Action on Elementary Tensors
Applying the left side of the rule to an elementary tensor first applies the maps T₁ and T₂ to the two components, then applies S₁ and S₂ to the results; applying the right side directly composes each pair of maps first and then applies the composed maps to the two components, producing the identical elementary tensor by either route.
Extension to General Tensors
Since both sides of the rule agree on every elementary tensor and both extend linearly across the tensor product space, they must agree on every general tensor as well, following the same uniqueness of linear extension used throughout the study of tensor products of maps.
Diagram of the Composition Rule
Two Routes to the Same Result
The diagram below shows composing the combined operators directly, compared with composing the factor maps first and then forming the tensor product of the results.
Special Cases of the Rule
One Factor Held Fixed at the Identity
When one of the four factor maps is the identity, the rule reduces immediately to the earlier composition compatibility result, since composing any map with the identity in the same slot leaves that map unchanged.
Composing an Operator With Its Own Inverse
If S₁ is the inverse of T₁ and S₂ is the inverse of T₂, the rule shows that composing the two combined operators produces the identity on the tensor product space, confirming the earlier result on invertibility of combined operators built from invertible factor maps.
Iterated Composition
Chains of Composed Combined Operators
Applying the composition rule repeatedly shows that composing any finite chain of combined operators built from the same two factor spaces reduces to a single combined operator formed from composing the corresponding chains of factor maps independently in each factor.
Order Preservation Within Each Factor
While the composition rule allows the overall chain to be decomposed factor by factor, the order of composition within each individual factor's chain must match the order of composition in the original chain of combined operators, since composition of operators is generally not commutative.
Matrix-Level Statement
Kronecker Product Compatibility With Matrix Products
Relative to fixed bases, the composition rule corresponds to the identity that the product of two Kronecker product matrices equals the Kronecker product of the corresponding matrix products, provided the matrices involved are conformable for multiplication in each factor.
Computational Advantage of the Rule
This matrix identity is frequently used to avoid ever constructing the large composite matrices directly, since the right-hand side only requires multiplying the smaller factor matrices before forming a single Kronecker product, rather than forming two large Kronecker products before multiplying them.
Extension to Several Factors
Composition Rule Across Many Factors
When the tensor product involves three or more factor spaces, the composition rule extends directly: composing two combined operators built from the same list of factor spaces reduces to composing the corresponding factor maps independently in every slot.
Compatibility With the Rest of the Framework
The composition rule works consistently alongside associativity, identity preservation, and the domain and codomain functoriality results, so a complex expression involving several combined operators, some fixed factors, and various compositions can always be simplified down to individual factor-level computations.