14.16.4 Tensor Map Product Linear Extension Evaluation

Tensor Map Product Linear Extension Evaluation explains how tensor maps interact via linear extensions in algebraic contexts.

Tensor Map Product Linear Extension Evaluation is the specific computational procedure for evaluating a combined operator on a general tensor by first writing that tensor as a finite sum of elementary tensors, applying the elementary evaluation rule to each term, and summing the resulting images, thereby carrying out in practice the linear extension that gives the combined operator its full domain.

The Procedure Step by Step

Decomposing the Input Into Elementary Terms

The first step of the procedure expresses the general input tensor as a finite sum of elementary tensors, a representation that always exists because elementary tensors span the tensor product space.

x = \sum_{k}^{} u_{k} \otimes v_{k}

Evaluating Term by Term and Summing

The second step applies the combined operator to each elementary term individually, using the elementary evaluation rule, and adds the resulting images together to obtain the final output.

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

Independence From the Chosen Decomposition

Multiple Valid Decompositions of the Same Tensor

A general tensor typically admits more than one valid decomposition into a sum of elementary tensors, since the number of terms and the specific vectors involved are not unique, only their combined tensor being fixed.

Guaranteed Agreement Across Decompositions

Regardless of which valid decomposition is chosen for the input tensor, applying the linear extension evaluation procedure produces exactly the same final output, a guarantee that follows from the well-definedness of the linear extension used to construct the combined operator in the first place.

Diagram of the Evaluation Procedure

From a Chosen Decomposition to a Guaranteed Output

The diagram below shows two different decompositions of the same input tensor both leading, through the linear extension evaluation procedure, to the same final output.

Practical Considerations for the Procedure

Choosing an Efficient Decomposition

While any valid decomposition produces the correct final output, choosing a decomposition with fewer elementary terms, or terms built from a convenient basis, typically reduces the amount of computation required to carry out the evaluation.

Using a Basis Decomposition

A particularly convenient choice of decomposition expresses the input tensor directly in terms of the induced basis of the tensor product space, since the coefficients relative to that basis are exactly the coordinates used in the matrix representation of the combined operator.

Relationship to the Matrix Representation

Matching the Coordinate-Based Evaluation

Carrying out the linear extension evaluation using a basis decomposition and then collecting the resulting coefficients reproduces exactly the same output as multiplying the coordinate column of the input tensor by the Kronecker product matrix representing the combined operator.

[(T_{1} \otimes T_{2}) (x)] = ([T_{1}] \otimes [T_{2}]) [x]

Two Equivalent Descriptions of the Same Computation

The linear extension evaluation procedure and the coordinate-based matrix multiplication are two different descriptions of the same underlying computation, one phrased directly in terms of vectors and elementary tensors, the other phrased in terms of coordinate columns and matrices.

Robustness of the Procedure Under Substitution

Substituting a Different Factor Map

If a factor map is replaced by a different map on the same factor space, the linear extension evaluation procedure remains valid without modification, since the decomposition of the input tensor does not depend on which factor maps are being applied.

Reusing a Decomposition Across Multiple Evaluations

Because the decomposition step depends only on the input tensor and not on the specific combined operator being applied, the same decomposition can be reused to evaluate several different combined operators on the same input tensor, avoiding repeated decomposition work.

Extension to Several Factors

Linear Extension Evaluation With Many Factors

When the tensor product involves three or more factor spaces, the same two-step procedure applies: the input tensor is decomposed into a sum of elementary tensors built from one vector per factor, and the combined operator is evaluated term by term using the multi-factor elementary evaluation rule before summing the results.

Consistency Across Different Numbers of Factors

The guarantee that the final output does not depend on the chosen decomposition holds regardless of how many factor spaces are involved, since it follows from the same underlying well-definedness of the linear extension that applies uniformly to tensor products of any number of factors.