# 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 $\map {\mathcal M_S} {m, n}$.

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 $S^{m \times n}$