14.22.5 Tensor Map Product Kronecker Notation

The Kronecker notation represents tensor map products, combining tensors through a structured operation that encodes multi-linear relationships in a compact matrix form.

Tensor Map Product Kronecker Notation is the specific display conventions used when writing out the matrix of $f \otimes g$ explicitly as a block matrix, together with the auxiliary vectorization notation used to relate Kronecker products to ordinary matrix equations.

Explicit Block Matrix Display

The Standard Block Layout

For $F$ an $n \times n$ matrix with entries $F_{i a}$ , the Kronecker product $F \otimes G$ is displayed explicitly as the block matrix

F \otimes G = [\begin{matrix} F_{11} G & \dots & F_{1 n} G \\ ⋮ & ⋱ & ⋮ \\ F_{n 1} G & \dots & F_{n n} G \end{matrix}]

with each block a scaled copy of the entire matrix $G$ , and the block in row $i$ , column $a$ of this outer array equal to the scalar $F_{i a}$ times $G$ .

Distinguishing Outer and Inner Indices Visually

Because this display nests one matrix inside each entry of another, notation for the Kronecker product typically uses a distinguishing typographic device, such as separating the outer index by a comma or writing the outer matrix in a different weight or size, to prevent a reader from mistaking the block structure for an ordinary matrix of scalars.

The Vectorization Operator

Definition of vec

The vectorization operator $vec$ stacks the columns of a matrix into a single column vector, so that for an $m \times n$ matrix $X$ , $vec (X)$ is the length- $m n$ column vector obtained by placing the first column of $X$ on top, the second column beneath it, and so on.

The Standard Identity Linking vec and the Kronecker Product

The vectorization operator and the Kronecker product are linked by the identity

vec (A X B) = (B^{T} \otimes A) vec (X)

which rewrites the matrix equation $A X B$ as an ordinary matrix-vector product against the stacked, vectorized form of $X$ , using the Kronecker product of $A$ with the transpose of $B$ .

Reading the Identity in Terms of the Tensor Map Product

This identity is the matrix-computation restatement of the tensor map product acting on a general element of a tensor product space, described in the tensor map product component summation case, since a matrix $X$ viewed as an element of $V \otimes W$ corresponds to $vec (X)$ under a fixed choice of basis input elements, and the left and right matrix multiplications by $A$ and $B$ correspond to the action of a tensor product map on that element.

Notational Care With Transposes in the Vec Identity

Why a Transpose Appears

The appearance of $B^{T}$ rather than $B$ itself in the vec identity is a direct consequence of stacking columns rather than rows; using a row-stacking convention for vectorization instead would move the transpose to the other factor, so any use of Kronecker notation together with vec must state explicitly which stacking convention, column-major or row-major, is in force.

Consistency With the Ordering Convention

This choice interacts directly with the tensor map product basis ordering convention discussed elsewhere, since the specific matrix produced by $vec$ depends on the same lexicographic choice that governs how basis input elements are linearized into a single index; changing one convention without correspondingly adjusting the other produces a mismatched, incorrect formula.

Practical Use in Notation

Compressing Matrix Equations for Computation

The vec-Kronecker identity is primarily a notational and computational convenience, allowing equations involving matrix multiplication on both sides of an unknown matrix to be rewritten as a single ordinary linear system in the unknown's vectorized form, a rewriting used extensively in numerical linear algebra software where only vector-based solvers may be directly available.