Definition:Vectorization of Matrix/Definition 1

Definition
Let $S$ be a set.

Let $m, n \ge 1$ be natural numbers.

Let $A = \sqbrk {a_{i j} }$ be a $m \times n$ matrix over $S$.

The vectorization of $A$ is the $m n \times 1$ column matrix:
 * $\map {\operatorname {vec} } A = \sqbrk {a_{1 1}, \ldots, a_{m 1}, a_{1 2}, \ldots, a_{m 2}, \ldots, a_{1 n}, \ldots, a_{m n} }^\intercal$

informally obtained by stacking the columns of $A$.

That is:
 * $\map {\operatorname {vec} } A_k = a_{\floor {k / m}, k \bmod m}$

where:
 * $\floor {\, \cdot \,}$ is the floor function
 * $\bmod$ is the modulo operation.

Also see

 * Equivalence of Definitions of Vectorization of Matrix