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

Proof

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Also see

• Vectorization of Product of Two Matrices