Permutation Matrix

Definition

$\Pi ={\prod }_{l=1}^k{\Pi }_{i_l{j}_l}$ where ${\Pi }_{ij}$ is a matrixed $i$-th and $j$-th rows are interchanged from an identity matrix $I$.

Facts

${\Pi }_{ij}^2=I$

$\Pi {\Pi }^{⊺}=I$