Is there a standard operation to "rotate rings on matrices"? Is there some standard operation to "rotate rings on matrices"? Look at the image below:

The numbers around the four empty squares are what I'm calling ring, In the second matrix, this ring has been rotated counterclockwise. I'm aware that ring may not be the right name.
 A: I'm pretty sure this operation has absolutely no interesting interpretation when the matrix is viewed as representing a linear transformation, as matrices usually are. It is therefore a fairly certain bet that this operation has no name.
A: Use the vector representation $\mathrm{vec}$ of your matrix $X$, by stacking all columns on top of each other:
$$
\mathrm{vec}\:
\pmatrix{
1 &4&2&6\\
7&\square&\square&2 \\
7&\square&\square&4 \\
8&2&5&6\\
}
=\pmatrix{
1 &4&2&6&
7&\square&\square&2 &
7&\square&\square&4 &
8&2&5&6\\
}^T.
$$
Now apply $\pi_{\text{rot.ring}}$, a permutation (matrix $M$, with dimension $4^2$) on $\mathrm{vec}\;X$, such that $\left(\mathrm{vec}\;X\right)_1$ is sent to $\left(\mathrm{vec}\;X\right)_5$, $\left(\mathrm{vec}\;X\right)_2$  to $\left(\mathrm{vec}\;X\right)_1$,...
So $\pi_{\text{rot.ring}}=\left(5,1,2,3,4,8,12,16,15,14,13,9\right)$ and $M_{j,j+1}=\left(\pi_{\text{rot.ring}}\right)_{j,j+1}$ and $M_{jj}=1$ if $j\in\{6,7,10,11\}$.
Undo the  $\mathrm{vec}$ operation and you'll get a matrix with your ring rotated.
A: If you identify the space $M_{n\times n}$ with $\mathbb R^{n^2}$, and let $S_{n^2}$ act on $\mathbb R^{n^2}$ by permuting the axes, then your operation corresponds to the action of a $4n-4$ cycle on $M_{n\times n}$. This interpretation is purely geometrical though, and does not give any understanding of how the operation affects the linear transformation corresponding to a matrix. 
