14.16 Tensor Map Product Evaluation Rule
The Tensor Map Product Evaluation Rule defines how tensor maps combine via multiplication in multilinear algebra.
Tensor Map Product Evaluation Rule is the procedure that specifies exactly how to compute the output of a combined operator built from a tensor product of maps when it is applied to a specific input tensor, reducing the evaluation to applying each individual factor map to its corresponding component and reassembling the results.
Evaluation on a Single Elementary Tensor
The Basic Evaluation Step
Given an elementary tensor formed from one vector per factor space, the combined operator is evaluated by applying each factor map to its own corresponding vector and combining the resulting images into a new elementary tensor.
No Interaction Required Between Factors
Because each factor map only needs its own corresponding vector as input, evaluating on an elementary tensor requires no communication or interaction between the computations happening in different factors, allowing them to proceed independently and in any order.
Evaluation on a General Tensor
Reducing to a Sum of Elementary Evaluations
A general tensor is a finite sum of elementary tensors, so evaluating the combined operator on a general tensor is carried out by evaluating on each elementary tensor in the sum individually and adding the resulting images together.
Well-Definedness Across Different Representations
Because a general tensor can be written as a sum of elementary tensors in more than one way, the evaluation rule must give the same final image regardless of which representation is used, a guarantee provided by the linear extension underlying the construction of the combined operator.
Diagram of the Evaluation Procedure
From Input Tensor to Output Tensor
The diagram below shows an input tensor being broken into elementary pieces, evaluated piece by piece, and reassembled into the final output tensor.
Evaluation via the Matrix Representation
Coordinate-Based Evaluation
Once bases are fixed for the factor spaces, evaluation can instead be carried out numerically by expressing the input tensor as a coordinate column relative to the induced basis and multiplying by the Kronecker product matrix representing the combined operator.
Equivalence of the Two Evaluation Methods
The coordinate-based evaluation and the direct elementary-tensor evaluation always agree, since the matrix representation is constructed precisely to reproduce the action of the combined operator on the induced basis, which underlies every general tensor's coordinate expansion.
Efficient Evaluation Without Forming the Full Matrix
Avoiding the Composite Matrix Entirely
For large tensor product spaces, evaluation can be performed without ever constructing the full Kronecker product matrix, by reshaping the coordinate representation of the input tensor and applying each factor matrix separately to the appropriate dimension.
Sequential Application to Each Mode
This more efficient evaluation strategy applies the first factor matrix along the dimension corresponding to the first factor space, then applies the second factor matrix along the dimension corresponding to the second factor space, producing the same final result as the full matrix multiplication while avoiding the multiplicative blow-up in size.
Evaluation With Special Factor Maps
Evaluation With an Identity Factor
If one factor map is the identity, evaluation simplifies immediately, since the corresponding component of every elementary tensor passes through unchanged, leaving only the nontrivial factor map's action to compute.
Evaluation With a Zero Factor
If one factor map is the zero map, evaluation on any tensor produces the zero tensor, since every elementary tensor in the sum has its corresponding component sent to zero, collapsing the entire sum to zero as well.
Extension to Several Factors
Evaluation With Many Simultaneous Factor Maps
When the combined operator is built from three or more factor maps, evaluation on an elementary tensor applies every individual factor map to its own corresponding vector simultaneously, exactly as in the two-factor case, with the results recombined into a single elementary tensor of the codomain space.
Extension to General Tensors With Several Factors
As with two factors, evaluation on a general tensor built from several factors reduces to evaluating on each elementary tensor in its sum representation and adding the resulting images, relying on the same linear extension and well-definedness guarantees established for the two-factor case.