14.9.2 Tensor Kronecker Operator Representation

The Tensor Kronecker Operator represents tensor products via matrix operations, combining multilinear maps in algebra.

Tensor Kronecker Operator Representation is the identification of a linear operator acting on a tensor product space with a single matrix built from the Kronecker product of the matrices representing simpler operators acting on the individual factor spaces. It provides the bridge between abstract linear maps defined on a tensor product and the concrete numerical arrays used to compute with them.

From Operators on Factors to a Single Operator

Combining Two Linear Maps

Given a linear map on the first factor space and a linear map on the second factor space, the operator representation constructs a single linear map on the tensor product space by acting with the first map on the first component and the second map on the second component of every simple tensor.

(S \otimes T) (u \otimes v) = S (u) \otimes T (v)

Matrix of the Combined Operator

Once bases are fixed for both factor spaces, the matrix of the combined operator with respect to the induced basis of the tensor product space is exactly the Kronecker product of the matrix of the first operator with the matrix of the second operator.

[S \otimes T] = [S] \otimes [T]

Representation Diagram

Operator Action Split Across Factors

The diagram below shows an operator on a tensor product space decomposing into independent action on each factor, then recombining into a single output vector in the product space.

Preservation of Algebraic Structure

Composition of Operator Representations

The Kronecker operator representation turns composition of operators on each factor into composition of the combined operator, so applying one combined operator after another matches applying the combined operator built from the composed factor operators.

(S_{1} S_{2}) \otimes (T_{1} T_{2}) = (S_{1} \otimes T_{1}) (S_{2} \otimes T_{2})

Identity and Inverse Operators

The combined operator built from two identity operators is the identity operator on the tensor product space, and when both factor operators are invertible, the combined operator is invertible with inverse equal to the Kronecker product of the individual inverses.

(S \otimes T)^{- 1} = S^{- 1} \otimes T^{- 1}

Eigenstructure Under the Representation

Eigenvalues of the Combined Operator

If a vector is an eigenvector of the first operator and another vector is an eigenvector of the second operator, their simple tensor is an eigenvector of the combined operator, with eigenvalue equal to the product of the two individual eigenvalues.

Full Eigenvalue Set

The complete set of eigenvalues of the combined operator consists of every pairwise product of an eigenvalue of the first operator with an eigenvalue of the second operator, so the eigenvalue structure of the combined operator is fully determined by the eigenvalue structures of its two factors.

Extension to Several Operators

Representation for More Than Two Factors

When the tensor product involves three or more factor spaces, the operator representation extends by taking the Kronecker product of the matrices of an operator on each individual factor, in the same order as the factors appear in the tensor product.

Consistency With Partial Application

Applying only some of the factor operators while leaving the remaining factors acted on by the identity operator produces a combined operator that acts nontrivially only on the corresponding subset of the tensor product space, leaving the other factors unchanged.