Matrix (correct) notation Say I have a real $m \times n$ matrix $\mathbf{M}$. Shall I write $\mathbf{M} \in \mathbb{R}^{m \times n}$ or $\mathbf{M} \in \mathbb{R}^{m,n}$?
What is commonly accepted and most beautiful and correct version?
Moreover, shall I use an italic bold font like $\boldsymbol{M}$ or vertical bold $\mathbf{M}$?  And same question for vectors: I personally dislike $\mathbf{x}$ and prefer $\boldsymbol{x}$. What is commonly and beautifully accepted?

In conclusion, if I got it right, $\boldsymbol{M}\in\mathbb{R}^{m \times n}$ and  $M_{i,j}\in\mathbb{R}$.
 A: I see this very often and was taught this:
$$M_{m,n}(\mathbb{R})$$
If it's just a square matrix, then
$$M_n(\mathbb{F})$$
where $\mathbb{F}$ is an arbitrary field.
These are the sets of matrices.  For an arbitrary matrix, say like your example.  Then I would see
$$\mathbf{A}\in{M_{m,n}(\mathbb{F})}$$
where $\mathbf{A}$ is your matrix.  I like the bold personally, but it is not required I don't believe.
A: The usual notation is $M\in\Bbb R^{m\times n}$ or $M\in {\mathrm M}_{m\times n}(\Bbb R)$. The bold face or italics is up to you, and depends on the context of what you're writing, and if you want to distinguish certain elements by such details. The "beautiful" part is simply subjective. I prefer $\bf x,y$ for vectors, and not-so-italicized capitals for matrices. In particular, there is no "correct" notation, but of course you won't be denoting matrices by the same symbols you denote vectors or scalars, say. The usual convention is upper case first or close-to-m alphabet letters for matrices, $A,B,C,M,N,\ldots$, lower-case for vectors, $a,b,v,w,x,y,\ldots$ and greek for scalars, $\lambda,\eta,\ldots$.
