14.4.4 Tensor Linear Operator Product Elementary Action

The Tensor Linear Operator Product Elementary Action describes how linear operators act on tensor spaces through fundamental algebraic operations.

Tensor Linear Operator Product Elementary Action is the rule specifying how the induced operator f tensor g acts on an elementary tensor v tensor w of the domain space, applying f to the first component and g to the second component independently and recombining the two results into a new elementary tensor.

Statement of the Elementary Action

The Defining Formula

For operators f on V and g on W, the elementary action is

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

a specialization to operators of the general elementary output rule for tensor products of maps, with the added feature that both sides of the formula now lie in the same space, V tensor W, since domain and codomain coincide for each of f and g.

Independence of the Two Component Actions

The elementary action treats the two components independently: f acts only on v, contributing no dependence on w, and g acts only on w, contributing no dependence on v, so the elementary action can be understood as applying two separate transformations, one to each component of the elementary tensor, and then reassembling them via the tensor product in the target space.

Building Up from the Elementary Action

Extension by Linearity to the Whole Domain Space

Since the domain space is spanned by elementary tensors, the elementary action determines the full action of f tensor g on every element of V tensor W by linear extension,

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

so knowledge of the elementary action alone is sufficient to determine f tensor g as an operator on the entire domain space, without needing to separately specify its behavior on non-elementary elements.

Coordinate Description of the Elementary Action

Given bases e-1 through e-m of V and h-1 through h-n of W, the elementary action on basis elementary tensors gives

(f \otimes g) (e_{i} \otimes h_{j}) = (\sum_{k} a_{ki} e_{k}) \otimes (\sum_{l} b_{lj} h_{l}),

where a and b are the matrices of f and g respectively, giving the exact coefficients of the elementary action directly in terms of the matrix entries of the two operators.

Consequences of the Elementary Action

Source of the Kronecker Product Representation

The block structure of the Kronecker product matrix A tensor B arises entirely from the elementary action applied to every pair of basis vectors, since the column of A tensor B corresponding to e-i tensor h-j is exactly the coordinate vector of the elementary action's output on that basis elementary tensor, arranged according to the ordering of the induced basis.

Source of the Product Eigenvalue Rule

If u is an eigenvector of f with eigenvalue lambda and x is an eigenvector of g with eigenvalue mu, the elementary action gives

(f \otimes g) (u \otimes x) = (λ u) \otimes (μ x) = λ μ (u \otimes x),

showing directly from the elementary action that u tensor x is an eigenvector of f tensor g with eigenvalue lambda times mu, the elementary action being the single computational step from which this entire spectral fact follows.