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 $\MM$.
Also see
- Results about matrix spaces can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: $2$: Some examples of rings: Ring Example $7$