14.7.3 Tensor Map Product Domain Element Form

Understanding how tensor map products operate on domain elements within algebraic structures.

Tensor Map Product Domain Element Form is the distinction between an element of the domain tensor space that is a single elementary tensor and one that is a general finite sum of elementary tensors, together with the consequences this distinction carries for evaluating a tensor product of maps.

The Two Basic Forms

Elementary Tensor Form

An element of the domain tensor space V1 tensor V2 is said to be in elementary tensor form when it is written as a single term

v \otimes w,

with v drawn from V1 and w drawn from V2, and elements of this form are exactly the ones on which the elementary output rule of a tensor product of maps is directly stated without any need for a further sum.

General Sum Form

An element not expressible as a single elementary tensor is said to be in general sum form,

\sum_{i = 1}^{k} v_{i} \otimes w_{i},

requiring more than one elementary tensor term to represent, and elements of this form require the extension of the elementary output rule by linearity before a tensor product of maps can be evaluated on them.

Determining the Form of a Given Element

Rank as a Measure of Element Form

The minimal number of terms k needed to express a given element of the domain tensor space in general sum form is called its rank as a tensor, and an element is in elementary tensor form precisely when its rank equals one or zero, the zero element being expressible with no terms at all.

Non-Uniqueness of the General Sum Form

Even fixing the rank, a general sum form is typically not unique: the same element of the domain tensor space can be written as a sum of the minimal number of elementary tensors in more than one way, so the domain element form specifies how many terms are needed but not a single canonical choice of those terms.

Consequences of the Element Form for Evaluation

Evaluation on Elementary Tensor Form

Evaluating a tensor product of maps on an element in elementary tensor form is immediate, requiring only a single application of the elementary output rule,

(f \otimes g) (v \otimes w) = f (v) \otimes g (w),

with no summation involved.

Evaluation on General Sum Form

Evaluating a tensor product of maps on an element in general sum form requires applying the elementary output rule to each term separately and summing the results,

(f \otimes g) (\sum_{i} v_{i} \otimes w_{i}) = \sum_{i} f (v_{i}) \otimes g (w_{i}),

with the same output guaranteed regardless of which particular decomposition into elementary tensors is used, by the well-definedness of the tensor product of maps construction.

The Element Form in Coordinate Computations

Coordinate Vectors of Each Form

An element in elementary tensor form has a coordinate vector, with respect to the induced basis, equal to the Kronecker product of the coordinate vector of v and the coordinate vector of w, a special structured vector; an element in general sum form has a coordinate vector equal to the sum of such Kronecker product vectors, one for each term in the sum, and this sum need not itself be expressible as a single Kronecker product unless the rank of the element happens to be one.

Recognizing Elementary Tensor Form from Coordinates

Given only a coordinate vector with respect to the induced basis, determining whether it represents an element in elementary tensor form amounts to checking whether the corresponding coefficient matrix, obtained by reshaping the coordinate vector according to the two factor dimensions, has rank at most one, connecting the domain element form directly to the ordinary matrix rank of this reshaped array.