Vectorization (mathematics)

The vectorization of a matrix is a linear transformation which converts the matrix into a column vector.
Specifically, the vectorization of $m \times n$ matrix $A$ , denoted $v ec (A)$ , is the $mn \times 1$ column vector obtained by stacking the columns of the matrix $A$ on top of one another.
- $vec (A) = [a_{1, 1}, \dots, a_{m, 1}, a_{1, 2}, \dots, a_{m, 2}, \dots, a_{1, n}, \dots, a_{m, n}]^{T}$
  - Here, $a_{i, j}$ represents $A (i, j)$

B) Example

for the $2 \times 2$ matrix $A = [a c b d]$ , the vectorization is $vec (A) = a c b d$

The vectorization is frequently used together with the Kronecker product to express matrix multiplication as a linear transformation on matrices.
$vec (A B) = (I_{m} \otimes A) vec (B) = (B^{T} \otimes I_{k}) vec (A)$
- $A$ and $B$ of dimensions $k \times l$ and $l \times m$ , respectively.

Vectorization is a unitary transformation from the space of $n \times n$ matrices with the Frobenius (or Hilbert–Schmidt) inner product to $C^{n^{2}}$ .
$tr (A^{⊤} B) = vec (A)^{⊤} vec (B) = vec (B)^{⊤} vec (A)$
- 쉽게 말해서 $tr (A^{⊤} B)$ 의 경우, 같은 위치에 해당하는 두 matrix 의 원소를 각기 서로 곱한다음 다 더한 값이 된다: $\sum a_{i, j} \cdot b_{i, j}$