# What is the notation for the set of all $m\times n$ matrices?

Given that $\mathbb{R}^n$ is the notation used for n-dimensional vectors, is there an accepted equivalent notation for matrices?

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If $A$ is an $m\times n$ matrix, then $$A \in \mathbb{R}^{m\times n}$$

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Since $\times$ is the official symbol for multiplying integers, this notation is in principle ambiguous: $2\times3=6$ would seem to imply $\Bbb R^{2\times 3}=\Bbb R^6$, but the latter is not a space of matrices. I agree that this is nitpicking, and would apply to the term "$m\times n$ matrix" itself too, but I personally prefer notations which use a comma rather than $\times$ (or nothing) between $m$ and $n$ (and this applies to matrix subscripts as well). – Marc van Leeuwen Oct 29 '13 at 10:30
I agree. If you just wrote $A\in \mathbb{R}^{m\times n}$ it wouldn't be necessarily clear $A$ is a matrix. – dezign Aug 20 '15 at 5:25

As you can see from the previous two answers, several notations are in common usage, so it's best to say what you mean the first time you use it to be completely clear. (For what it's worth, I often use $M_{m\times n}(\mathbb{R})$, which is different again).

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Does your notation refer to $m \times n$ or to $n \times m$ matrices? (The OP asks for $m \times n$.) – Ryan Oct 29 '13 at 4:31
@Ryan Thanks - force of habit. – Matthew Pressland Oct 29 '13 at 9:44
Np, I asked because my textbook just did the same thing, and it got me confused (thus ending up here after searching for the answer). – Ryan Oct 29 '13 at 13:10

The vector space of real matrices with $n$ rows and $m$ columns is denoted by $\mathcal{M}_{n,m}(\mathbb R)$ and its $nm$-dimensional vector space so it's isomorphic to $\mathbb R^{nm}$

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Oh, I did not pay attention to what the OP wrote and yet I think my answer does not make confusion. – user63181 Oct 29 '13 at 14:03

The notation $A\in\mathbb{R}^{m\times n}$ is in fact correct. We should be careful with the symbol '$\times$' that in this case does not means an integer product but a group product that, as a result, is a new group in $\mathbb{R}^2$ with elements of the form $(m, n)$, and thus the matrix $A$ is contained in it.

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