14.9 Tensor Kronecker Product Relation

Tensor Kronecker Product Relation is the precise correspondence linking the abstract tensor product of linear maps to the Kronecker product of matrices, establishing that the two notions coincide exactly once bases are fixed and the appropriate induced basis of elementary tensors is used on each tensor product space.

Statement of the Relation

The Correspondence Itself

For linear maps f and g represented, with respect to chosen bases, by matrices A and B, the Kronecker product relation states

where square brackets denote the matrix of a map with respect to the relevant basis, and the right side is the ordinary Kronecker product of the two matrices A and B.

The Bases Required for the Relation to Hold

The relation holds specifically with respect to the induced bases of elementary tensors on the domain and codomain tensor spaces, built from the chosen bases of the individual factor spaces in a fixed order; using any other basis of the tensor product spaces, not built this way from the factor bases, would generally require a change-of-basis correction before the relation as stated applies.

Verification of the Relation

Agreement on Basis Elementary Tensors

The relation is verified by checking that both sides act identically on every basis elementary tensor e-i tensor h-j of the domain: the left side gives f tensor g applied directly, while the right side gives the corresponding column of the Kronecker product matrix, and both are shown, by direct computation with the elementary output rule, to produce the same coordinate vector.

Extension by Linearity

Since both sides of the relation are linear maps agreeing on a spanning set, namely the basis elementary tensors, the two sides agree everywhere on the domain tensor space, establishing the relation as an identity of linear maps rather than merely an identity checked on a limited set of test vectors.

Consequences of the Relation

Translation of Algebraic Identities

The Kronecker product relation allows every algebraic identity established for tensor products of maps to be translated directly into a matrix identity for Kronecker products, and conversely: the composition identity

corresponds, via the relation, exactly to the mixed product property of Kronecker products.

Invariant Formulas Preserved by the Relation

Because the relation identifies f tensor g with A tensor B entrywise, every invariant computable from a matrix, such as trace, determinant, rank, and eigenvalues, computed on A tensor B, matches exactly the corresponding invariant of the abstract map f tensor g, so the relation guarantees that abstract and matrix computations of these invariants always agree.

Limits of the Relation

Dependence on Basis Choice

The Kronecker product relation, while always available once bases are fixed, depends on the specific bases chosen for the factor spaces: a different choice of bases for V1, V2, W1, or W2 produces a different pair of matrices A and B, and hence a different Kronecker product A tensor B, representing the same abstract map f tensor g but with respect to a different induced basis of the tensor product spaces.

Relation to Change of Basis

If P and Q are change-of-basis matrices for the domain factor spaces, and R and S are change-of-basis matrices for the codomain factor spaces, the Kronecker product relation interacts with change of basis through

confirming that the Kronecker product of the transformed matrices matches the Kronecker product relation applied to the individually transformed matrices of f and g, so the relation is stable under any consistent change of basis on each factor.