Is there a special notation for sets of matrices?

Question

So, I know that you can write $\mathbb{R}^n$ to express the set of all n-tuples with real components. Is there a similar notation for the set of all $m\times n$ matrices, something like $\mathbb{R}^{m \times n}$?

Answer 1

$\mathbb R^{m \times n}$, $M_{m \times n}(\mathbb R)$, $\mathcal M_{m \times n} (\mathbb R)$, and $\mathrm{Mat}_{m \times n}(\mathbb R)$ are all fairly standard notations for the set of $m \times n$ real matrices. $m \times n$ in the subscripts can also be replaced with something like $m,n$, to give $M_{m,n}(\mathbb R)$. Also common is $\mathcal M(m,n)$ (where $\mathbb R$ is assumed), and if the matrices are square then the double dimensions are often dropped, so $M_{n}(\mathbb R)$ is the set of all real $n \times n$ matrices.