14.2.4 Tensor Kronecker Product Area

The Tensor Kronecker Product Area explores how tensor products combine multidimensional arrays to model complex linear relationships in algebra and beyond.

Tensor Kronecker Product Area is the branch of study concerned with the matrix representation of tensor products of linear maps, in which the induced map on a tensor product of finite-dimensional spaces is expressed concretely as a Kronecker product of the matrices representing the individual maps.

From Abstract Map to Concrete Matrix

Setting Up Bases

Given linear maps

f : V_{1} \to W_{1}

g : V_{2} \to W_{2}

represented, with respect to chosen bases, by matrices A of size m by n and B of size p by q, the Kronecker product area asks how the abstract tensor product map f tensor g should be represented once bases for V1 tensor V2 and W1 tensor W2 are built from the elementary tensors of the original basis vectors.

The Kronecker Product Formula

The answer is that f tensor g is represented, in this induced basis, by the block matrix

A \otimes B = (\begin{matrix} a_{11} B & \dots & a_{1n} B \\ ⋮ & ⋱ & ⋮ \\ a_{m1} B & \dots & a_{mn} B \end{matrix}),

an m p by n q matrix built by replacing every entry of A with that entry times the entire matrix B, so that the Kronecker product converts the abstract tensoring of maps into an explicit block construction on ordinary matrices.

Algebraic Identities of the Kronecker Product

Mixed Product Property

The central computational identity of this area is the mixed product property,

(A \otimes B) (C \otimes D) = (AC) \otimes (BD),

valid whenever the ordinary matrix products AC and BD are defined, and this identity is the matrix-level counterpart of the composition rule for tensor products of maps.

Transpose, Inverse, and Determinant

The Kronecker product interacts predictably with the standard matrix operations:

{(A \otimes B)}^{T} = A^{T} \otimes B^{T},

{(A \otimes B)}^{-1} = A^{-1} \otimes B^{-1},

the second holding whenever A and B are square and invertible, and

det (A \otimes B) = det (A)^{q} \cdot det (B)^{p},

when A is p by p and B is q by q, giving direct formulas for these invariants without recomputing them from the full block matrix.

Spectral and Rank Computations

Eigenvalues via the Kronecker Product

For square matrices A and B, if lambda is an eigenvalue of A and mu is an eigenvalue of B, then lambda times mu is an eigenvalue of A tensor B, and the full spectrum of A tensor B, with multiplicity, is exactly the multiset of all such pairwise products, a fact used to compute spectra of large structured matrices from the much smaller spectra of A and B.

Rank Multiplicativity

The rank of the Kronecker product satisfies

rank (A \otimes B) = rank (A) \cdot rank (B),

which is verified by reducing A and B to their row echelon forms and observing that the Kronecker product of the reduced forms retains a block structure whose nonzero blocks are in exact bijection with the products of nonzero pivot positions of A and B.

Computational Advantages

Reduced Storage and the Vec Identity

Because A tensor B is fully determined by the much smaller matrices A and B, problems involving A tensor B can often be solved without ever forming the large matrix explicitly. This is exploited through the identity

vec (BXC) = (C^{T} \otimes B) vec (X),

where vec stacks the columns of a matrix into a single vector, allowing linear matrix equations involving Kronecker products to be rewritten as equations directly in terms of B, C, and X, avoiding the explicit formation of the larger Kronecker product matrix.