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.