14.15.3 Tensor Map Product Factorwise Composition

Tensor Map Product Factorwise Composition combines tensor maps by applying them sequentially to tensor factors, preserving structure across multi-linear operations.

Tensor Map Product Factorwise Composition is the practice of computing the composition of two combined operators by working entirely within the individual factor spaces, composing the corresponding pair of maps in each factor separately and only forming the tensor product of the results at the very end, rather than composing the full combined operators directly.

The Factorwise Procedure

Splitting the Task Into Two Smaller Compositions

Given two combined operators to compose, the factorwise procedure identifies the pair of maps acting on the first factor and the pair of maps acting on the second factor, composing each pair independently before recombining.

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

Justification From the Composition Rule

This procedure is justified directly by the general composition rule for tensor products of maps, which guarantees that computing the composition this way produces exactly the same combined operator as composing the two full combined operators directly.

Advantages of Working Factorwise

Reduced Problem Size

Because each individual factor space is typically much smaller than the full tensor product space, composing the pair of maps within each factor involves working with much smaller objects than composing the full combined operators directly.

Reusable Intermediate Results

If several different combined operators share the same map on one factor, the composition involving that shared factor map can be computed once and reused across multiple factorwise compositions, avoiding redundant work.

Diagram of the Factorwise Approach

Composing Small Pieces Before Recombining

The diagram below shows two combined operators being composed by first working within each factor separately, then assembling the two factor-level results into the final combined operator.

Applying Factorwise Composition to Longer Chains

Extending the Procedure to Several Composed Operators

For a chain of several combined operators built from the same two factor spaces, the factorwise procedure composes the entire chain of maps within the first factor, composes the entire chain of maps within the second factor, and only forms the tensor product of the two resulting composed maps at the final step.

Preserving the Chain Order Within Each Factor

As with any factorwise computation, the order of composition within each factor's chain must exactly match the order of the original chain of combined operators, since reversing the order within a factor generally changes the result.

Factorwise Composition With Identity or Invertible Factors

Simplification When a Factor Chain Reduces to the Identity

If the chain of maps within one factor composes down to the identity, the factorwise procedure shows that the entire combined operator reduces to acting only through the other factor, since the identity contributes nothing to that factor's component.

Factorwise Verification of Inverses

Factorwise composition provides a direct way to verify that a proposed combined operator is the inverse of another: composing the corresponding factor maps in each slot and checking that each factor-level composition yields the identity on that factor confirms that the full combined operators are inverses of one another.

(T_{1} \circ T_{1}^{- 1}) \otimes (T_{2} \circ T_{2}^{- 1}) = I

Matrix-Level Factorwise Composition

Multiplying Small Factor Matrices First

At the matrix level, factorwise composition corresponds to multiplying the smaller factor matrices in each slot first, and only forming the Kronecker product of the resulting matrix products at the final step, matching the composition rule expressed in matrix form.

(A \otimes B) (C \otimes D) = (A C) \otimes (B D)

Practical Computational Savings

This matrix-level version of factorwise composition avoids ever forming the full composite matrices at intermediate stages, computing the two smaller matrix products separately and forming the single large Kronecker product only once, at the very end of the computation.

Extension to Several Factors

Factorwise Composition Across Many Factors

When the tensor product involves three or more factor spaces, factorwise composition extends by composing the corresponding chain of maps within every individual factor separately, and only assembling the final combined operator by taking the tensor product of all the resulting factor-level compositions.

Independence of the Per-Factor Computations

Because the composition within each factor is computed entirely independently of the others, the individual factor-level compositions in a multi-factor setting can be carried out in any order, or even simultaneously, without affecting the final combined operator obtained by recombining them.