TITLE: Is there any standard notation for specifying dimension of a matrix after the matrix symbol?
QUESTION [7 upvotes]: I want to explicitly specify dimension of matrices in some expressions, something like 
$$\boldsymbol{A}_{m \times n} \boldsymbol{B}_{n \times m} = \boldsymbol{C}_{m \times m} \, .$$
Is there any more or less standard notation for this?

While this notation is generally unambiguous, I think it become ambiguous, for example, in the following case. Suppose that I have some column or row vector which is conjugate transpose of correspondingly row or column vector. If I write it in the following way 
$$\boldsymbol{A}_{n \times 1}^{\dagger} \, ,$$
we can have 2 different interpretations:

matrix $\boldsymbol{A}_{n \times 1}$ is $n \times 1$ column vector and by
$\boldsymbol{A}_{n \times 1}^{\dagger}$ I'm referring to its
conjugate transpose which is $1 \times n$ row vector;
matrix $\boldsymbol{A}_{n \times 1}^{\dagger}$ itself is $n \times 1$ column vector which is conjugate transpose of $1 \times n$ row vector $\boldsymbol{A}$.

REPLY [2 votes]: As requested, the subscript $m \times n$ notation is perfectly common, if not standard.
Of course, we can always use parenthesis to resolve any difficulties. It should be clear that $$\left(A_{m\times n}\right)^T = \left(A^T\right)_{n\times m}.$$
Of course, with a simple operation like the transpose, it looks much nicer as just $A^T$, or, if necessary $\left(A_{m\times n}\right)^T$. The dimensionality of the resulting transpose should be obvious.
Nevertheless, it's not bad practice especially for teaching to remove any possible ambiguities when doing several matrix-matrix or matrix-vector multiplications.