# 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 $\map {\MM_S} n$.

### Real Matrix Space

Let $\R$ denote the set of real numbers.

The $m \times n$ matrix space over $\R$ is referred to as the real matrix space, and can be denoted $\map {\MM_\R} {m, n}$.

### Complex Matrix Space

Let $\C$ denote the set of complex numbers.

The $m \times n$ matrix space over $\C$ is referred to as the complex matrix space, and can be denoted $\map {\MM_\C} {m, n}$.

## 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 $\MM$.