Vectorization (matrix, $\mathrm{vec}(A)$)

Column-stacking vectorization turns an $m\times n$ matrix into a vector of length $mn$. For $X=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$, $\mathrm{vec}(X)=(a,c,b,d{)}^{\top }$. A row-stacking convention gives a different ordering and must not be silently substituted.

For compatible matrices, column stacking giveswhere $\otimes$ 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.