Definition:Matrix Space

Definition
Let $m, n \in \Z_{>0}$ be (strictly) positive integers.

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 $\map {\MM_S} {m, n}$.

Thus, by definition:
 * $\map {\MM_S} {m, n} = S^{\closedint 1 m \times \closedint 1 n}$

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

Also denoted as
Various forms of $\MM$ may be used; $\mathbf M$ and $M$ being common.

Some sources denote $\map {\MM_S} {m, n}$ as:


 * $\map {\mathbf M_{m, n} } S$
 * $S^{m \times n}$

Similarly, $\map {\MM_S} n$ can be seen as:


 * $\map {\mathbf M_n} S$
 * $S^{n \times n}$

with varying styles of $\MM$.

Also see

 * Ring of Square Matrices over Ring is Ring