14.14.2 Tensor Map Product Composition Preservation

Tensor Map Product Composition Preservation ensures that the product of tensor maps respects composition, maintaining structural integrity across transformations.

Tensor Map Product Composition Preservation is the specific half of the functorial behavior of the tensor product construction stating that forming the tensor product of two composed maps in each factor equals the composition of the tensor products taken factor by factor, one of the two conditions required for the tensor product to qualify as a functor.

Precise Statement

Two-Factor Statement

Given two maps composed on the first factor space and two maps composed on the second factor space, the tensor product of the two compositions equals the composition of the tensor products of the corresponding pairs of maps.

(S_{1} \circ T_{1}) \otimes (S_{2} \circ T_{2}) = (S_{1} \otimes S_{2}) \circ (T_{1} \otimes T_{2})

Direct Verification on Elementary Tensors

Applying the left side to an elementary tensor composes each factor's two maps before applying them to the corresponding vector, while applying the right side applies the inner tensor product first and the outer tensor product second; both routes apply the same sequence of individual maps to the same individual vectors, producing the same elementary tensor as a result.

[(S_{1} \circ T_{1}) \otimes (S_{2} \circ T_{2})] (u \otimes v) = S_{1} (T_{1} (u)) \otimes S_{2} (T_{2} (v))

Why This Counts as Functorial

The Second of the Two Functor Conditions

A functor must preserve composition of morphisms in addition to preserving identity morphisms; composition preservation addresses exactly this second requirement for the tensor product construction viewed as a functor of two variables.

Independence From Identity Preservation

Composition preservation is a separate condition from identity preservation and must be established on its own terms, even though both properties together are what justify calling the tensor product construction functorial rather than merely a convenient notation.

Diagram of Composition Preservation

Two Equal Paths Through Composition and Tensoring

The diagram below shows composing first and tensoring second on one path, and tensoring first and composing second on the other path, both reaching the same final combined operator.

Consequences of Composition Preservation

Decomposing a Combined Operator's Composition

Whenever a computation involves composing two combined operators, composition preservation permits that composition to be carried out factor by factor instead, often reducing the size of the intermediate objects that must be computed.

Iterated Composition

Applying composition preservation repeatedly shows that composing any finite chain of combined operators equals the combined operator formed from composing the corresponding chains of maps in each factor independently.

(S_{1} \otimes S_{2}) \circ (R_{1} \otimes R_{2}) \circ (T_{1} \otimes T_{2}) = (S_{1} \circ R_{1} \circ T_{1}) \otimes (S_{2} \circ R_{2} \circ T_{2})

Matrix-Level Confirmation

Kronecker Product Compatibility With Matrix Multiplication

Relative to fixed bases, composition preservation corresponds to the identity that the Kronecker product of two matrix products equals the product of the corresponding Kronecker products, provided the factor matrices are conformable for multiplication in each slot.

(A_{1} B_{1}) \otimes (A_{2} B_{2}) = (A_{1} \otimes A_{2}) (B_{1} \otimes B_{2})

Computational Advantage

This matrix identity allows a computation to choose whichever side of the equality is cheaper to evaluate, often preferring to multiply the smaller factor matrices first and take a single Kronecker product afterward, rather than forming two large Kronecker products before multiplying them.

Extension to Several Factors

Composition Preservation Across Many Factors

When the tensor product involves three or more factor spaces, composition preservation holds independently in every factor slot, so composing combined operators built from several factors reduces to composing the corresponding factor maps slot by slot.

Compatibility With Associativity and Identity Preservation

Composition preservation works alongside the associativity of the tensor product and identity preservation, allowing any expression combining these three properties to be simplified in whichever order is most convenient, always reaching the same final combined operator.