14.13 Tensor Map Product Associativity Behavior

Tensor Map Product Associativity Behavior is the property that combining three or more linear maps by repeated tensor product produces the same resulting operator regardless of how the maps are grouped in pairs before combining, so that the tensor product of maps requires no fixed grouping convention to be unambiguous.

Statement of the Associativity Property

Two Equivalent Groupings

Given three linear maps, grouping the first two together before combining with the third produces the same combined operator as grouping the last two together before combining with the first.

Underlying Reason for the Property

This behavior mirrors the associativity of the tensor product of vector spaces themselves: since the underlying spaces combine associatively up to a natural identification, the maps acting on those spaces combine associatively in the same way, acting identically on corresponding elementary tensors under either grouping.

Associativity Diagram

Two Paths to the Same Combined Operator

The diagram below shows three individual maps reaching the same final combined operator through two different grouping paths.

Consequence for Notation

Justification for Omitting Parentheses

Because the two possible groupings of three factors always produce the same result, the tensor product of three or more maps can be written without parentheses, using a single unambiguous expression that lists all the factors in order.

Extension to Any Number of Groupings

For four or more factors, every possible way of inserting parentheses to group the factors in pairs before combining produces the same final combined operator, so the tensor product of any finite list of maps is well defined without reference to any particular grouping.

Associativity at the Level of Elementary Tensors

Action on a Triple Elementary Tensor

Applying the combined operator built under either grouping to an elementary tensor formed from three vectors, one from each factor space, produces the same result: each individual map acts on its own corresponding vector, and the three images are combined into a single elementary tensor of the codomain space.

Extension to General Tensors by Linearity

Since general tensors are finite sums of elementary tensors, and the linear extension of an operator does not depend on how the underlying tensor product space was grouped when the extension was defined, associativity of the combined operator's action on general tensors follows directly from its associativity on elementary tensors.

Compatibility With the Matrix Representation

Associativity of the Kronecker Product

At the level of matrices, this associativity behavior corresponds exactly to the associativity of the Kronecker product: forming the Kronecker product of the first two factor matrices before combining with the third produces the same composite matrix as forming the Kronecker product of the last two factor matrices before combining with the first.

Independence of Computational Strategy

Because associativity holds exactly, a computation involving a long tensor product of maps can be carried out by combining factors in whatever order is most convenient computationally, without affecting the final result.

Associativity Combined With Other Properties

Interaction With Composition

Associativity of the tensor product of maps holds alongside, and independently of, the earlier composition compatibility rule that governs how tensor products of maps interact with ordinary composition of operators; the two properties can be applied together in any combination when simplifying a long expression involving several tensor products and compositions.

Interaction With Identity Factors

Associativity behavior applies equally when some of the factors involved are identity maps, so grouping a chain of maps that includes identity factors in different ways still produces the same final combined operator, consistent with the earlier discussion of identity map compatibility.