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

Also see

• Ring of Square Matrices over Ring

Sources

• Seth Warner: Modern Algebra (1965): $\S 29$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.2$: Ring Example $7$