14.19.3 Tensor Map Product Operator Matrix Transformation

The Tensor Map Product Operator transforms matrices through algebraic operations, enabling complex tensor interactions in mathematical structures.

Tensor Map Product Operator Matrix Transformation is the special case of the transformation behavior of $f \otimes g$ that applies when $f$ and $g$ are operators, meaning endomorphisms $T : V \to V$ and $S : W \to W$ on the same space in both domain and codomain, so that the matrix of the combined operator $T \otimes S$ transforms by an ordinary similarity transformation rather than by the more general two-sided change used when domain and codomain bases can differ.

Reduction to a Single Similarity Transformation

Matching Domain and Codomain Bases

When $T$ and $S$ are operators, the same basis is used for domain and codomain of each factor, so a single change of basis matrix $P$ on $V$ and $Q$ on $W$ governs the transformation of both matrices,

\tilde{T} = P T P^{- 1}, \tilde{S} = Q S Q^{- 1}

and the matrix of the combined operator transforms as the single similarity transformation

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

Consequence for Similarity Invariants

Since $T \otimes S$ is genuinely conjugated, rather than transformed by unrelated matrices on each side, every similarity invariant of the matrix, its trace, determinant, characteristic polynomial, rank, and eigenvalues with multiplicity, is unaffected by this transformation, exactly as for any single operator undergoing a change of basis.

Eigenvalue Behavior Under the Transformation

Eigenvalues of the Combined Operator

If $λ$ is an eigenvalue of $T$ with eigenvector $v$ , and $μ$ is an eigenvalue of $S$ with eigenvector $w$ , then $v \otimes w$ is an eigenvector of $T \otimes S$ with eigenvalue $λ μ$ ,

(T \otimes S) (v \otimes w) = T (v) \otimes S (w) = λ v \otimes μ w = λ μ (v \otimes w)

so that the full spectrum of $T \otimes S$ , counted with multiplicity, is exactly the set of pairwise products of eigenvalues of $T$ and eigenvalues of $S$ .

Invariance of This Fact Under the Matrix Transformation

This spectral product structure survives any similarity transformation of $T$ and $S$ individually, since eigenvalues of a matrix are unchanged by conjugation; the eigenvectors themselves transform along with the change of basis, becoming $P v \otimes Q w$ in the new coordinates, but the eigenvalue $λ μ$ attached to that eigenvector is unaffected by which basis is used to express it.

Diagonalization Behavior

Simultaneous Diagonalization From Separate Diagonalizations

If $T$ is diagonalized by a change of basis $P$ , so that $P T P^{- 1}$ is diagonal, and $S$ is diagonalized by $Q$ , so that $Q S Q^{- 1}$ is diagonal, then the operator matrix transformation by $P \otimes Q$ diagonalizes $T \otimes S$ as well, since the Kronecker product of two diagonal matrices is diagonal, with diagonal entries equal to all pairwise products of the two sets of eigenvalues.

Non-Diagonalizable Factors

If either $T$ or $S$ fails to be diagonalizable, no choice of $P$ and $Q$ of the restricted Kronecker form $P \otimes Q$ can diagonalize $T \otimes S$ , though a more general similarity transformation not of Kronecker form may still succeed in special cases where the Jordan structure of the two factors happens to combine into a diagonalizable result.

Restriction Compared to the General Case

What Is Lost by Restricting to Operators

The general basis change response of a tensor product map allows four independent matrices, two for the domain side and two, generally unrelated, for the codomain side; restricting to operators collapses this to two matrices by forcing the codomain change to equal the domain change on each factor. This restriction is precisely why operator matrix transformation is describable as a similarity transformation, while the transformation of a general map $f : V \to V^{'}$ with $V \neq V^{'}$ is not, in general, a similarity transformation at all.