Vectorization of Product of Three Matrices

Theorem
Let $R$ be a ring.

Let $A, B, C$ be matrices over $R$ such that the matrix product $ABC$ is defined.

Then $\map {\operatorname {vec} }{ABC} = \paren {C^\intercal \otimes A} \cdot \map {\operatorname {vec} } B$ where:
 * $\operatorname {vec}$ denotes vectorization
 * $C^\intercal$ is the transpose of $C$
 * $\otimes$ denotes Kronecker product
 * $\cdot$ denotes matrix product

Also see

 * Vectorization of Product of Two Matrices