14.11.4 Tensor Operator Product Elementary Tensor Action

Exploring how elementary tensors interact via operator products in algebraic frameworks and their structural implications.

Tensor Operator Product Elementary Tensor Action is the rule describing exactly how a combined operator built from individual factor operators acts on a single elementary tensor, that is, a simple tensor formed from one vector taken from each factor space, by applying each factor operator to its own corresponding vector independently and then re-forming the elementary tensor from the results.

The Action Rule

Applying Factor Operators Component by Component

Given an elementary tensor built from one vector per factor space, the combined operator acts by applying the first factor operator to the first vector, the second factor operator to the second vector, and so on, then combining the resulting vectors back into a single elementary tensor.

(T_{1} \otimes T_{2}) (u \otimes v) = T_{1} (u) \otimes T_{2} (v)

Defining Property of the Construction

This action on elementary tensors is not merely a consequence of the combined operator's construction; it is the defining condition used to construct the combined operator in the first place, since the combined operator is defined by specifying its behavior on every elementary tensor and then extending that behavior linearly.

Diagram of the Elementary Action

Applying Two Operators in Parallel

The diagram below shows an elementary tensor being acted on by two factor operators simultaneously, one per component, producing a new elementary tensor.

Extension From Elementary Tensors to General Tensors

Linearity Extends the Action

Since every tensor in the product space is a finite sum of elementary tensors, the action of the combined operator on a general tensor is obtained by applying the elementary action to each elementary tensor in the sum and adding the results, using the linearity of the combined operator.

(T_{1} \otimes T_{2}) (\sum_{k}^{} u_{k} \otimes v_{k}) = \sum_{k}^{} T_{1} (u_{k}) \otimes T_{2} (v_{k})

Well-Definedness Across Different Representations

Because a general tensor may be written as a sum of elementary tensors in more than one way, the elementary tensor action rule must produce the same final result regardless of which representation is used, a property guaranteed by the universal property underlying the tensor product construction.

Behavior on Basis Elementary Tensors

Action on Basis Pairs

When the vectors in the elementary tensor are basis vectors of their respective factor spaces, the combined operator's action reduces to combining the images of those basis vectors under the individual factor operators, which are themselves expressed as linear combinations of the basis vectors of the codomain spaces.

Recovering the Matrix Representation

Recording the action of the combined operator on every basis elementary tensor, and expressing each result in the induced basis of the codomain tensor space, reproduces exactly the columns of the Kronecker product matrix representing the combined operator.

Special Cases of the Elementary Action

Identity Action on a Factor

If one factor operator is the identity, the elementary tensor action leaves the corresponding component of the elementary tensor completely unchanged, transforming only the components associated with the other, nontrivial factor operators.

Zero Action on a Factor

If one factor operator sends its input vector to the zero vector, the entire resulting elementary tensor becomes the zero tensor, since an elementary tensor with any zero component is itself the zero tensor.

Extension to Several Factors

Elementary Action With Many Factor Operators

When the combined operator involves three or more factor operators, the elementary tensor action rule applies each factor operator to its corresponding component of the elementary tensor simultaneously, exactly as in the two-factor case, with no interaction required between the different components.

Order Independence of the Component-Wise Application

Since each factor operator acts only on its own designated component of the elementary tensor, the order in which the individual component actions are carried out does not affect the final resulting elementary tensor.