Definition:Matrix Space/Complex

Definition

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

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 $\map {\MM_\C} {m, n}$.

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

Also denoted as

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

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

• $\map {M_{m, n} } \C$
• $\C^{m \times n}$

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

• $\map {M_n} \C$
• $\C^{n \times n}$

with varying styles of $\MM$.

Sources

• 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.1$ Matrices