14.16.1 Tensor Map Product Elementary Input
The Tensor Map Product Elementary Input explains how tensor maps combine through basic operations, forming foundational algebraic structures in tensor algebras.
Tensor Map Product Elementary Input is the specific case of input to a combined operator that consists of a single elementary tensor, formed from exactly one vector taken from each factor space, representing the simplest and most direct case for which the evaluation rule of a tensor product of maps can be applied without any need to first decompose the input into a sum of simpler pieces.
Characterizing an Elementary Input
Structure of the Input
An elementary input to a combined operator consists of one vector chosen from each factor space, joined together by the tensor product symbol to form a single tensor in the product space.
Distinction From a General Tensor
An elementary input differs from a general tensor in the product space in that it requires no sum of multiple terms to express; a general tensor may require several elementary tensors added together, while an elementary input is a single such term standing alone.
Evaluating on an Elementary Input
Direct Application of Each Factor Map
Because the input is already in elementary form, the combined operator can be evaluated immediately by applying each factor map to its own corresponding vector, without any preliminary decomposition step.
Simplicity as the Basis for the General Evaluation Rule
Because every general tensor decomposes into a sum of elementary tensors, the evaluation rule for elementary input serves as the fundamental building block from which the evaluation rule for arbitrary tensors is constructed, using linear extension to handle the additional terms of a sum.
Diagram of an Elementary Input
One Vector Per Factor Joined Into a Single Tensor
The diagram below shows two vectors, one from each factor space, joined together to form a single elementary input tensor.
Elementary Input Expressed in Coordinates
Coordinate Vector as a Tensor Product of Coordinate Vectors
Once bases are fixed for the factor spaces, the coordinate column of an elementary input relative to the induced basis of the tensor product space is itself the Kronecker product of the coordinate column of the first vector and the coordinate column of the second vector.
Reduced Computational Cost for Elementary Input
Because the coordinate column of an elementary input is itself a Kronecker product of two much smaller vectors, evaluating a combined operator on an elementary input can be computed by applying each factor matrix separately to the corresponding smaller coordinate vector, without ever assembling the full coordinate column of the elementary input or the full composite matrix.
Recognizing Elementary Inputs Within a General Computation
Identifying an Elementary Structure
Before applying a general evaluation procedure, it is often useful to check whether a given input tensor happens to be elementary, since recognizing this structure allows the more efficient elementary evaluation to be used directly rather than falling back to a full decomposition into a sum of several terms.
Not Every Tensor Is Elementary
Many tensors in a product space of dimension greater than the sum of the factor dimensions cannot be written as a single elementary tensor at all, and require a genuine sum of two or more elementary tensors to express, so the elementary input case, while foundational, does not cover every possible input to a combined operator.
Elementary Input With Special Factor Maps
Identity Applied to an Elementary Input
If one factor map is the identity, evaluating on an elementary input leaves the corresponding vector completely unchanged while the other vector is transformed by its own nontrivial factor map, producing a new elementary tensor with only one altered component.
Zero Map Applied to an Elementary Input
If one factor map is the zero map, evaluating a combined operator on any elementary input produces the zero tensor, since the corresponding component vanishes and an elementary tensor with a zero component is itself the zero tensor.
Extension to Several Factors
Elementary Input With Many Factor Spaces
When the tensor product involves three or more factor spaces, an elementary input consists of one vector chosen from each individual factor space, and evaluation proceeds by applying every factor map to its own corresponding vector simultaneously, exactly as in the two-factor case.
Role as the Foundation for General Tensors With Several Factors
Just as with two factors, every general tensor built from three or more factor spaces decomposes into a sum of elementary inputs, so the elementary evaluation rule established here remains the essential foundation for evaluating combined operators on arbitrary multi-factor tensors.