Mathematical notation of matrix row or column circular permutation

Question

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

Answer 1

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.)

Answer 2

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.