14.15.5 Tensor Map Product Composition Result Preservation

Tensor Map Product Composition Preserves Results through Algebraic Structure and Tensor Algebra Properties.

Tensor Map Product Composition Result Preservation is the property that the specific properties possessed by the composition of two factor maps in each individual factor, such as invertibility, rank, or being an isomorphism, carry through unchanged into the corresponding property of the composed combined operator, once the composition rule has been applied to reduce the combined composition to its factorwise pieces.

What Is Being Preserved

Properties of the Factorwise Compositions

Once the composition rule reduces the composition of two combined operators to a tensor product of two factor-level compositions, any property already established for those two factor-level compositions is a property of the pieces from which the combined operator is built.

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

From Factor-Level Property to Combined Property

Because the right-hand side of the composition rule is itself a tensor product of the two factor-level compositions, the structure transport principle applies directly, carrying properties of these compositions through to the resulting combined operator.

Preservation of Invertibility

Invertible Factorwise Compositions

If the composition on the first factor, S₁ composed with T₁, is invertible, and the composition on the second factor, S₂ composed with T₂, is invertible, then the composed combined operator is invertible, with inverse given by the tensor product of the two individual inverses.

[(S_{1} \otimes S_{2}) \circ (T_{1} \otimes T_{2})]^{- 1} = (S_{1} \circ T_{1})^{- 1} \otimes (S_{2} \circ T_{2})^{- 1}

Failure of Invertibility in a Single Factor

If either factor-level composition fails to be invertible, the composed combined operator is not invertible, since a noninvertible operator in either factor produces a nontrivial kernel that extends into a nontrivial kernel of the full combined operator.

Diagram of Result Preservation

Factor-Level Properties Determining the Combined Result

The diagram below shows properties of the two factor-level compositions flowing into the corresponding property of the composed combined operator.

Preservation of Rank

Multiplicative Rank Relationship

The rank of the composed combined operator equals the product of the rank of the first factor's composition and the rank of the second factor's composition, following the same multiplicative rule that governs the rank of any tensor product of two linear maps.

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

Rank Reduction Through Composition

Since composing two operators generally cannot increase rank beyond the smaller of the two individual ranks, this multiplicative relationship shows how a reduction in rank within either factor's composition directly reduces the rank of the entire combined operator.

Preservation for Isomorphisms

Composed Isomorphisms Remain Isomorphisms

If the composition in each factor is itself an isomorphism between the relevant spaces, the composed combined operator is an isomorphism between the corresponding tensor product spaces, since an isomorphism is exactly an invertible linear map, and invertibility has already been shown to transfer through the construction.

Isomorphism Class Determined Factor by Factor

The specific isomorphism realized by the combined operator is completely determined by the two individual factor-level isomorphisms, so studying the combined isomorphism reduces entirely to studying its two simpler factor-level components.

Extension to Several Factors

Result Preservation Across Many Factors

When the tensor product involves three or more factor spaces, composition result preservation extends directly: any property held by every individual factor-level composition, such as invertibility or a specific rank value, determines the corresponding property of the fully composed combined operator across all factors.

Combining With Other Established Rules

Composition result preservation works alongside factorwise composition and composable factor maps, allowing a complex chain of combined operators across several factors to be fully analyzed by reducing to individual factor-level compositions and reading off their properties directly.