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 $$\mathbf M_n \left({S}\right)$$