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Is this a valid notation for real $m \times n$ matrices: $\mathbb{R}^{m,n}$. $m$ and $n$ are known.

If it is not, then what would be the right notation for the set of such matrices?

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$\mathbb R^{m \times n}$ is sometimes used. This notation also nicely resembles the standard English phrase "$m \times n$ matrix" used to describe them. –  Srivatsan Sep 26 '11 at 20:11
    
I would say use $M^{m\times n}(\mathbb{R})$ to denote $m$ by $n$ matrices over reals. –  Sasha Sep 26 '11 at 20:13
    
I for one hate that notation :) I always use $M_{n,m}(\mathbb R)$. –  Mariano Suárez-Alvarez Sep 26 '11 at 20:13
    
Thank you all. If someone can post it as answer, I'll accept it. –  I J Sep 26 '11 at 20:26
    
@Mariano I posted an answer about the $\mathbb R^{m \times n}$ notation only. Perhaps you should add another answer with your favorite notation. :) –  Srivatsan Sep 26 '11 at 20:30
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3 Answers

up vote 2 down vote accepted

Some people use $\mathbb R^{m \times n}$ to denote $m \times n$ matrices over the reals. Though this notation is perhaps not standard, I like it because:

  1. It resembles the usual English phrase "$m \times n$ matrix of reals" used to describe these matrices. (Admittedly, the notation $M^{m \times n}(\mathbb R)$ suggested by Sasha conveys the same idea equally easily.)

  2. When $n$ is $1$ (i.e., for a column vector), the notation $\mathbb R^{m \times 1}$ is close to the standard notation $\mathbb R^m$ for column vectors, which is nice. On the other hand, the notation $\mathbb R^{1\times m}$ standing for a row vector looks equally similar to $\mathbb R^{1 \times m}$. So one must be careful if one distinguishes between row and column vectors.

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Usually, the denotation for row/column vector is saved by the convention of using boldface small letters for vectors, and boldface capitals for matrices. Witness $\mathbf x \in \mathbb R^{1\times p}$ and $\mathbf Q \in \mathbb C^{4\times 3}$... –  J. M. Sep 26 '11 at 21:17
    
@J.M. Ah yes. I didn't quite think about that. (And, in any case, I was arguing it's a good thing that the notation $\mathbb R^{m \times 1}$ looks similar to $\mathbb R^m$ and it's not so great that $\mathbb R^{1 \times m}$ also looks similar. I am wondering if I didn't convey that clearly ;)) –  Srivatsan Sep 26 '11 at 21:20
    
The other cure(?) is to do something like $\mathbf v^\top \in \mathbb C^{m\times 1}$. (Okay, slightly obtuse, that one...) –  J. M. Sep 26 '11 at 21:23
    
@J.M. Related to the last point, I wonder if people say $\mathbf v \in (\mathbb C)^{\ast}$ (meaning the dual space) though. –  Srivatsan Sep 26 '11 at 21:26
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I have seen many different notations in use but there is unfortunately no standard. I like to use $\mathcal{M}^m_n(\mathbb{F})$ to denote all matrices $m \times n$ over a given field $\mathbb{F}$. It is then consistent, for a given $A \in \mathcal{M}^m_n(\mathbb{F})$, to denote the element at row $i$ column $j$ by $a^i_j$. This representation (in many cases) facilitates the summation convention (when defining multiplication, expanding by bases, etc). Moreover, it is also natural to denote the $i^{th}$ row vector of $A$ by $A^i$ and the $j^{th}$ column vector of $A$ by $A_j$

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I actually like this definition. Surprising that it does not show up that much in linear algebra books. –  I J Sep 27 '11 at 21:36
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A similar notation, not mentioned above, is $\mathrm{Mat}_{m,n}(\mathbb{R})$. Also I suppose it should be mentioned that in the case $m = n$, one tends to only put in a simple sub- or superscript and write something like $\mathrm{Mat}_n(\mathbb{R})$, $M_n(\mathbb{R})$ or $\mathcal{M}_n(\mathbb{R})$.

In different contexts, other notations arise depending on the relevant structure on the space of matrices. For example, when talking about Lie algebras, another way of denoting the space of $n \times n$-matrices is $\mathfrak{gl}_n(\mathbb{R})$.

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