Vectorization

Definition

Let $A=[{\mathbf{a}}_1,{\mathbf{a}}_2,\dots ,{\mathbf{a}}_n]$ be an $m\times n$ matrix, then $\mathrm{vec}(A)=[a_{11},a_{21},\dots ,a_{m1},a_{12},a_{22},\dots ,a_{m2},\dots ,a_{1n},a_{2n},\dots ,a_{mn}{]}^{⊺}$

A linear transformation which converts the matrix into a vector.

Facts

Let $A,B,C,Z$ be matrices, then

$\mathrm{vec}(ABC)=(C^{⊺}\otimes A)\mathrm{vec}(B)$ $\mathrm{tr}(AB)=\mathrm{vec}(A^{⊺}{)}^{⊺}\mathrm{vec}(B)$ $\mathrm{tr}(A{Z}^{⊺}BZC)=\mathrm{vec}(Z^{⊺}{)}^{⊺}(CA\otimes B^{⊺})\mathrm{vec}(Z)$