Mathematical notation of matrix row or column circular permutation I'm having a trouble finding a proper mathematical notation for circular permutation of rows or columns of a matrix, for example: 
$$
A=
\begin{bmatrix}
a & b & c\\
d & e & f 
\end{bmatrix}
$$
If we apply a row circular permutation $$\sigma_{row}(A)$$ twice the results is: 
$$
\begin{bmatrix}
d & e & f \\
a & b & c
\end{bmatrix}
And
\begin{bmatrix}
a & b & c\\
d & e & f 
\end{bmatrix}
$$
If we apply a column circular permutation $$\sigma_{column}(A)$$ three times, the results is: 
$$
\begin{bmatrix}
c & a & b\\
f & d & e 
\end{bmatrix}
,
\begin{bmatrix}
a & b & c\\
d & e & f 
\end{bmatrix}
And
\begin{bmatrix}
b & c & a\\
e & f & d 
\end{bmatrix}
$$
Is there a notation for this, with details about its properties or special properties, what is its properties in case of convolution ?
Thanks
 A: Row permutations can be written as $g \to \sigma g$ for $g\in M_{n, m}$ and some fixed $\sigma\in M_n$. For example, 
\begin{align*}
\sigma_{row}(g) &= \pmatrix{0 & 1 \\ 1 & 0}g.
\end{align*}
In general, for a permutation $s\in S_n$, the matrix $\sigma$ with
\begin{align*}
\sigma_{ij} &= \delta_{s(i), j} = \begin{cases}
1 & \text{if $s(i) = j$;} \\
0 & \text{otherwise}
\end{cases}
\end{align*}
has
\begin{align*}
(\sigma g)_{ij} &= \sigma_{ik} g_{kj} = \delta_{s(i), k}\, g_{kj} = g_{s(i), j},
\end{align*}
as desired. Column permutations can be written as $g \to g\sigma$, either by the same argument by taking transposes.
A: Rows can be permuted by multiplying on the left by a permutation matrix; columns can be permuted by multiplying on the right by a permutation matrix. Explicitly for circular permutations:
$$
    \pmatrix{
        0 & 1 & 0 & 0 \\
        0 & 0 & 1 & 0 \\
        0 & 0 & 0 & 1 \\
        1 & 0 & 0 & 0 \\
    }
    \pmatrix{
        \mathrm{row}_1 \\
        \mathrm{row}_2 \\
        \mathrm{row}_3 \\
        \mathrm{row}_4 \\
    }
    =
    \pmatrix{
        \mathrm{row}_2 \\
        \mathrm{row}_3 \\
        \mathrm{row}_4 \\
        \mathrm{row}_1 \\
    }
$$
$$
    \pmatrix{
        \mathrm{col}_1
        & \mathrm{col}_2
        & \mathrm{col}_3
        & \mathrm{col}_4
    }
    \pmatrix{
        0 & 1 & 0 & 0 \\
        0 & 0 & 1 & 0 \\
        0 & 0 & 0 & 1 \\
        1 & 0 & 0 & 0 \\
    }
    =
    \pmatrix{
        \mathrm{col}_4
        & \mathrm{col}_1
        & \mathrm{col}_2
        & \mathrm{col}_3
    }
$$
To perform the inverse permutation, use the inverse matrix. In this example, it's
$$
    \pmatrix{
        0 & 0 & 0 & 1 \\
        1 & 0 & 0 & 0 \\
        0 & 1 & 0 & 0 \\
        0 & 0 & 1 & 0 \\
    }.
$$
The generalization to other dimensions should be obvious.
In other words, permuting rows and columns is a special case of matrix multiplication. The standard notation is to use ordinary matrix multiplication. So, write it as multiplying your matrix $A$ by certain specific matrices. (There is no standard notation for these specific matrices being used, though, nor is there a standard notation for the matrix corresponding to a permutation.)
