14.16.5 Tensor Map Product Evaluation Consistency

Tensor Map Product Evaluation Consistency is the property that every valid method of evaluating a combined operator on a given input tensor, whether by direct decomposition into elementary terms, by coordinate-based matrix multiplication, or by any other correct computational route, produces exactly the same output tensor, so the evaluation of a combined operator is a well-defined operation independent of the specific method used to carry it out.

Statement of Consistency

Agreement Across Methods

For a fixed combined operator and a fixed input tensor, the output obtained by decomposing the tensor into elementary terms and evaluating directly matches the output obtained by expressing everything in coordinates and multiplying by the Kronecker product matrix, which in turn matches the output obtained from any other valid evaluation route.

Sources That Feed Into Consistency

This consistency rests on several previously established facts working together: the well-definedness of the linear extension across different decompositions, the equivalence between elementary evaluation and coordinate-based evaluation, and the independence of factor evaluation from the order in which the individual factors are processed.

Consistency Across Different Decompositions

No Dependence on the Chosen Elementary Terms

As already established for linear extension evaluation, decomposing the same input tensor into two different sets of elementary terms and evaluating each decomposition separately always yields the same final output tensor, which is the most basic layer of evaluation consistency.

Consistency Under Reordering of Terms

Because addition of tensors is commutative, reordering the terms within a single decomposition before summing their evaluated images has no effect on the final output, adding a further layer of robustness to the consistency property.

Diagram of Consistency Across Methods

Multiple Paths Converging on One Output

The diagram below shows three different evaluation methods, elementary decomposition, coordinate matrix multiplication, and factorwise evaluation, all converging on the same output tensor.

Consistency Under Change of Basis

Coordinate Values Change, Output Vector Does Not

While the specific numerical coordinate column representing the output tensor depends on the basis chosen for the codomain tensor product space, the abstract output tensor itself, considered independently of coordinates, is exactly the same regardless of which basis was used to carry out the coordinate-based computation.

Consistent Translation Between Bases

Any two coordinate columns representing the same output tensor under different bases are related by the appropriate change of basis matrix, and applying this relation confirms that both coordinate computations describe the identical underlying output tensor.

Consistency Under Alternative Groupings

Independence From the Grouping of Factors

When more than two factors are involved, evaluation consistency also holds across the different possible groupings permitted by associativity: evaluating the combined operator built under one grouping produces the same output tensor as evaluating the combined operator built under any other grouping, once the natural identification between the differently grouped tensor product spaces is applied.

Consistency With Composition-Based Evaluation

If a combined operator is itself expressed as a composition of two simpler combined operators, evaluating it directly on an input tensor produces the same output as evaluating the two simpler combined operators in sequence, consistent with the general composition rule established for tensor products of maps.

Practical Significance of Consistency

Freedom to Choose the Most Convenient Method

Because every valid evaluation method is guaranteed to agree, a computation can freely choose whichever method is most efficient or most convenient for the specific situation at hand, without any risk of obtaining a different, incorrect answer by making that choice.

Basis for Verifying Implementations

Evaluation consistency provides a natural way to check a numerical implementation of the tensor product construction: computing the same evaluation through two different methods and confirming that the results agree serves as a direct test of correctness for the implementation.

Extension to Several Factors

Consistency Across Many Simultaneous Factors

When the tensor product involves three or more factor spaces, the same evaluation consistency holds: every valid method of computing the action of the combined operator on a given input tensor, whether by direct decomposition, coordinate multiplication, or factorwise evaluation extended across all factors, produces the identical output tensor.

Reliability as the Number of Factors Grows

This consistency does not weaken or become less reliable as the number of factors increases; the same guarantees that hold for two factors extend directly to any finite number of factors, since they rest on properties, such as linear extension and associativity, that themselves hold uniformly regardless of the number of factors involved.