Is There a Standard Notation for Matrix Rows and Columns?

[Max.Planck]

Is there notation to denote the i-th row of a Matrix or j-th column?

 

[Simon Bridge]

Yes - several. What was the context?
http://en.wikipedia.org/wiki/Matrix_(mathematics)#Notation

eg. if $a_{i,j}$ denotes the i/jth element of $\mathbf{A}$ then $a_{i,*}$ is a common way to denote the ith row of $\mathbf{A}$... or maybe the rows and columns would be represented as vectors.

 

[Max.Planck]

For example in a proof I would like to say:

$\exists i$ such that the i-th column of A = the j-th row of B, basically I'm looking for the notation of i-th column, and j-th row.

 

[Square]

I sometimes use the notation $A_{\bullet i}$ to denote the i-th column and $A_{j\bullet}$ to denote the j-th row.

 

[Max.Planck]

Is there a standard for notation?

 

[Simon Bridge]

Is there a standard for notation?

No - as the wikipedia article (see link above) points out there is no standard (i.e. ISO standard) notation. Some journals may specify a particular form in their style guides. afaik the main ISO standard for matrix notation specifies bold-face (poss bf-italics).

Per your example: one would most commonly write something like:$$\mathbf{A},\mathbf{B}\in \mathbb{M}_{m\times n}; a_{i,j}\in \mathbf{A}, b_{i,j}\in \mathbf{B}: a_{i,*}=b_{*,i}$$... and be fairly confident of being understood to mean that the ith row of A is the same as the ith column of B (pretty much the transpose but a fair example and you can have different functions of the row/column numbers.)