14.15.4 Tensor Map Product Composition Equality

Tensor Map Product Composition Equality explores when composing tensor maps, the order of multiplication affects the resulting tensor product's structure and properties.

Tensor Map Product Composition Equality is the precise statement, together with its justification, that the composition of two tensor products of maps is exactly equal, rather than merely similar or numerically close, to the tensor product of the corresponding factorwise compositions, established by verifying agreement on elementary tensors and extending that agreement to the whole space by linearity.

Statement of the Equality

The Equality Itself

For maps S₁, T₁ on the first factor and S₂, T₂ on the second factor, with the appropriate domains and codomains matching, the composition of the two combined operators equals the combined operator built from the two factorwise compositions.

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

What Exact Equality Means Here

This is an equality of functions between the same domain and codomain: for every possible input tensor, without exception, the two sides produce precisely the same output tensor, not merely outputs that agree up to some approximation or only for special choices of input.

Verification Strategy

Step One: Agreement on Elementary Tensors

The first step of the verification checks that both sides of the equality, applied to an arbitrary elementary tensor, produce the same result, by directly tracing through the definition of the tensor product of maps and the definition of composition.

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

Step Two: Extension by Linearity

Because elementary tensors span the entire tensor product space and both sides of the equality are linear maps, agreement on every elementary tensor forces agreement on every finite sum of elementary tensors, and therefore on every tensor in the space, completing the verification.

Diagram of the Verification Structure

Two Steps Leading to Full Equality

The diagram below shows the two-step verification process, moving from agreement on elementary tensors to agreement across the entire tensor product space.

Why the Two-Step Method Is Sufficient

Uniqueness of Linear Extensions

A linear map on the tensor product space is completely determined by its values on any spanning set of that space, and elementary tensors form such a spanning set; two linear maps that agree on a spanning set must agree everywhere, which is exactly why the two-step verification method is logically sufficient.

No Additional Cases to Check

Because the spanning property of elementary tensors covers every tensor in the space through finite linear combinations, there is no tensor left unaccounted for once the two-step verification is complete, so no further special cases need to be checked separately.

Consequences of Establishing the Equality

Basis for the General Composition Rule

This equality is precisely the general composition rule for tensor products of maps, and once established rigorously it can be applied freely in any computation involving composed combined operators, without needing to re-derive it from elementary tensors each time.

Foundation for Further Derived Results

Many later results, including the transport of invertibility, the multiplicative rank formula, and the preservation of composition results, rely directly on this composition equality as their starting point, making it a central fact within the broader study of tensor products of maps.

Extension to Several Factors

Composition Equality Across Many Factors

When the tensor product involves three or more factor spaces, the same two-step verification method establishes the corresponding composition equality: agreement on elementary tensors built from one vector per factor, followed by extension by linearity across sums of such elementary tensors.

Consistency With the Two-Factor Case

The multi-factor composition equality specializes exactly to the two-factor case when only two of the factors carry nontrivial maps and the rest are held at the identity, confirming that the more general statement is a genuine extension rather than a separate, unrelated result.