Definition:Matrix Space

Definition
Let $m, n \in \Z_+$, and let $S$ be a set.

The $m \times n$ matrix space over $S$ is defined as the set of all $m \times n$ matrices over $S$, and is denoted $\mathcal M_S \left({m, n}\right)$.

Thus, by definition:
 * $\mathcal M_S \left({m, n}\right) = S^{\left[{1 \, . \, . \, m}\right] \times \left[{1 \, . \, . \, n}\right]}$

If $m = n$ then we can write $\mathcal M_S \left({m, n}\right)$ as $\mathcal M_S \left({n}\right)$.

Alternative notation
Some sources denote:


 * $\mathcal M_S \left({m, n}\right)$ as $\mathbf M_{m,n} \left({S}\right)$


 * $\mathcal M_S \left({n}\right)$ as $\mathbf M_n \left({S}\right)$