---- > [!definition] Definition. ([[vectorize operation]]) > The **vectorize operation** $\text{vec}$ takes as input a [[matrix]] $\b X \in \mathbb{F}^{M \times N}$ and outputs a [[vector]] in $x \in \mathbb{F}^{MN}$ formed by stacking the $N$ columns of $\b X$ on top of each other. In other words for $i=qM+r$ with $0\leq r<M$ (see [[division algorithm]]), $[\text{vec}(\b X)]_{i}=\b X_{qM, r}=\b X_{qM, i \text{ mod }M}= \b X_{\lceil \frac{i}{M} \rceil, i \text{ mod }M}$ > while $\b X_{ij}=[\text{vec}(\b X)]_{M\cdot(j-1)+i} \ \ .$ ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```