= Vectorization
{disambiguate=matrix}
{title2=$\operatorname{vec}(A)$}
= Vectorization
{synonym}
Column-stacking <vectorization (matrix)> turns an $m\times n$ <matrix> into a vector of length $mn$. For $X=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, $\operatorname{vec}(X)=(a,c,b,d)^\top$. A row-stacking convention gives a different ordering and must not be silently substituted.
For compatible <matrices>, column stacking gives
$$
\operatorname{vec}(AXB)=(B^\top\otimes A)\operatorname{vec}(X),
$$
where $\otimes$ is the <Kronecker product>. Comparing the coefficient of each $X_{kl}$ on both sides proves the identity. The transpose of $B$ is ordinary transpose even over the complex numbers; <complex conjugation> enters only if the original expression itself uses an adjoint. This identity turns <matrix> equations into linear systems in their entries.
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