14.13.4 Tensor Map Product Reassociation Rule
The Tensor Map Product Reassociation Rule governs how tensor products interact under mappings, ensuring consistency in algebraic transformations across tensor spaces.
Tensor Map Product Reassociation Rule is the explicit procedure for converting a tensor product of maps expressed under one grouping of parentheses into an equivalent expression under a different grouping, relying on the underlying associativity of the tensor product to guarantee that the reassociated expression represents the same combined operator.
Purpose of the Rule
Moving Between Equivalent Groupings
Given a tensor product of several maps written with a specific placement of parentheses, the reassociation rule specifies exactly how to rewrite that expression with parentheses placed differently, while guaranteeing the resulting operator is unchanged.
Reliance on the Underlying Identification
The rule is justified by the natural identification between the two possible ways of parenthesizing the tensor product of the underlying vector spaces, which matches every elementary tensor formed one way to the corresponding elementary tensor formed the other way.
Applying the Rule Step by Step
Locating the Parentheses to Move
The first step in applying the rule is identifying the specific pair of parentheses in the current expression that should be shifted, based on which grouping is more convenient for the next stage of a computation.
Rewriting Without Changing the Factor Order
The reassociation rule only changes the placement of parentheses; it never changes the left-to-right order in which the individual factor maps appear, since reordering the factors is a separate operation governed by different rules.
Diagram of a Reassociation Step
Shifting a Single Pair of Parentheses
The diagram below shows a reassociation step applied to a product of three maps, moving the parentheses from grouping the first two factors to grouping the last two factors.
Reassociation for Longer Products
Reducing a Long Product to Repeated Triple Steps
For a tensor product of four or more maps, any desired reassociation can be achieved by a sequence of individual reassociation steps, each one only shifting a single pair of parentheses among three consecutive factors at a time, following the triple factor grouping rule repeatedly.
Order of Steps Does Not Affect the Final Outcome
Because every intermediate reassociation step preserves the identity of the combined operator, the final result of a sequence of reassociation steps depends only on the final placement of parentheses reached, not on the particular sequence of individual steps used to get there.
Reassociation Combined With Substitution
Reassociating Before Substituting a Factor
The reassociation rule can be applied before replacing one of the factor maps with a different map, changing which pair of factors is grouped together without affecting the validity of a subsequent substitution.
Reassociating After Simplifying a Subgroup
If a particular grouped pair of factors simplifies to a single operator, for instance because one of the two factors is the identity map, the reassociation rule can be applied first to bring those two factors together, allowing the simplification to be carried out before reassociating back if needed.
Matrix-Level Reassociation
Reassociating Kronecker Products
At the level of matrices, the reassociation rule corresponds to shifting the parentheses in a repeated Kronecker product of factor matrices, with the associativity of the Kronecker product guaranteeing that the resulting composite matrix is unaffected by the change in grouping.
Practical Use in Computation Planning
In practice, the reassociation rule is used to choose whichever grouping produces the smallest intermediate matrices during a computation, since forming the Kronecker product of two smaller factors first can reduce the memory required at each intermediate step, even though the final composite matrix is the same regardless of the grouping chosen.
Scope and Limits of the Rule
Reassociation Alone Does Not Reorder Factors
The reassociation rule strictly preserves the order of the factors; obtaining a version of the product with the factors listed in a different order requires a separate reordering argument, which does not generally hold for the tensor product of general operators in the same unconditional way that reassociation does.
Reassociation Applies Regardless of the Specific Factor Maps
The reassociation rule holds no matter what the individual factor maps are, whether they are invertible, singular, identity maps, or any other kind of linear map, since the underlying justification depends only on the associativity of the tensor product of the vector spaces themselves.