Definition:Vectorization of Matrix

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 $S$.

Definition 1
The vectorization of $A$ is the $m n \times 1$ column matrix:
 * $\map {\operatorname {vec} } A = \sqbrk {a_{11}, \ldots, a_{m1}, a_{12}, \ldots, a_{m2}, \ldots, a_{1n}, \ldots, a_{mn} }^\intercal$

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.

Definition 2
Let $R$ be a ring with unity.

Let $A$ be an $m\times n$ matrix over $R$.

The vectorization of $A$ is its coordinate vector with respect to the standard matrix basis.

Also see

 * Equivalence of Definitions of Vectorization of Matrix
 * Vectorization of Product of Three Matrices
 * Vectorization of Product of Two Matrices