Permutation Matrix

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Matrix

$\begin{bmatrix} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}$

Exactly one 1 in every row

Exactly one 1 in every column.

Everywhere else is 0.

Swapping Rows

$\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$ $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}=$ $\begin{bmatrix} d & e & f \\ a & b & c \\ g & h & i \\ \end{bmatrix}$

Swapping Columns

$ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix} $ $ \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}^\intercal= $ $ \begin{bmatrix} b & a & c \\ e & d & f \\ h & g & i \\ \end{bmatrix} $

Swapping Both

$\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$ $\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}$ $\begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}^\intercal$ $=\begin{bmatrix} e & d & f \\ b & a & c \\ h & g & i \\ \end{bmatrix} $

Full Example

$\begin{bmatrix} 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}$ $\begin{bmatrix} 0 & 1 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 0 \\ \end{bmatrix}$ $\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}$ $ = \begin{bmatrix} 0 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ \end{bmatrix} $