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:
 * $\map {\mathcal M_S} {m, n} = S^{\closedint 1 m \times \closedint 1 n}$

If $m = n$ then we can write $\map {\mathcal M_S} {m, n}$ as $\map {\mathcal M_S} n$.

Also denoted as
Some sources denote:


 * $\map {\mathcal M_S} {m, n}$ as $\map {\mathbf M_{m, n} } S$


 * $\map {\mathcal M_S} n$ as $\map {\mathbf M_n} S$


 * $\map {\mathcal M_S} {m, n}$ as $S^{m \times n}$

Also see

 * Ring of Square Matrices over Ring is Ring