Definition:Vectorization of Matrix

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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:

$\map {\operatorname {vec} } A = \sqbrk {a_{1 1}, \ldots, a_{m 1}, a_{1 2}, \ldots, a_{m 2}, \ldots, a_{1 n}, \ldots, a_{m n} }^\intercal$

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.


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