14.11.5 Tensor Operator Product Linear Extension

Tensor Operator Product Linear Extension extends tensor operations through linear algebra, blending structure and transformation in mathematical frameworks.

Tensor Operator Product Linear Extension is the process by which the action of a combined operator, initially defined only on elementary tensors formed from a pair of factor operators, is extended to act on every tensor in the product space by requiring the extended map to respect addition and scalar multiplication of tensors.

Necessity of the Extension

Elementary Tensors Do Not Exhaust the Space

The elementary tensors span the tensor product space but do not by themselves account for every element of it; a general tensor is a finite sum of elementary tensors, so an operator defined only on elementary tensors is not yet defined on the whole space until that definition is extended.

The Extension Requirement

To obtain an operator on the full tensor product space, the action already specified on elementary tensors must be extended so that the resulting map is linear, meaning it commutes with addition of tensors and with multiplication of a tensor by a scalar.

T (x + y) = T (x) + T (y)

T (c x) = c T (x)

Constructing the Extension

Extending Across a Sum of Elementary Tensors

Given a general tensor written as a finite sum of elementary tensors, the extended operator is defined by applying the elementary tensor action to each term of the sum individually and then adding the resulting terms together.

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

Well-Definedness of the Extension

Because a general tensor can be written as a sum of elementary tensors in more than one way, the extension is only meaningful if applying the rule to any two different representations of the same tensor produces the same final result; this well-definedness is guaranteed by the universal property that characterizes the tensor product construction.

Diagram of the Extension Process

From Elementary Action to Full Linear Map

The diagram below shows the elementary tensor action being extended, term by term, across a finite sum to produce the action on a general tensor.

Uniqueness of the Extension

Only One Linear Extension Exists

Given a fixed action on elementary tensors, there is exactly one linear map on the full tensor product space that agrees with that action, since any two linear maps that agree on a spanning set of a vector space must agree everywhere on that space.

Consequence for Defining Combined Operators

This uniqueness is what allows a combined operator to be defined simply by stating its action on elementary tensors, without separately verifying that a consistent linear map exists for every possible tensor in the space.

Compatibility With Other Structure

Extension Preserves Composition

The linear extension of a composition of factor operators agrees with the composition of the linear extensions of the individual combined operators, so extending first and composing afterward gives the same result as composing first and extending afterward.

Extension Preserves the Matrix Representation

Once bases are fixed, the linearly extended operator's matrix, expressed in the induced basis of the tensor product space, is exactly the Kronecker product of the matrices of the individual factor operators, confirming that the extension recovers the expected numerical representation.

Extension for Several Factors

Extending an Action Defined on Several Factors at Once

When the combined operator is built from three or more factor operators, the same extension process applies: the action is first specified on elementary tensors built from one vector per factor, and then extended linearly across sums of such elementary tensors to cover the entire multi-factor tensor product space.

Consistency Regardless of the Number of Factors

The uniqueness of the linear extension holds regardless of how many factor operators are combined, so a combined operator built from many factors is just as well-defined by its elementary tensor action as one built from only two factors.