Permutation Matrix

A Matrix ${\hbox{\sf p}}_{ij}$ obtained by permuting the $i$th and $j$th rows of the Identity Matrix with $i<j$. Every row and column therefore contain precisely a single 1, and every permutation corresponds to a unique permutation matrix. The matrix is nonsingular, so the Determinant is always Nonzero. It satisfies

$${{\hbox{\sf p}}_{ij}}^2={\hbox{\sf I}},$$

where I is the Identity Matrix. Applying to another Matrix, ${\hbox{\sf p}}_{ij}{\hbox{\sf A}}$ gives A with the $i$th and $j$th rows interchanged, and ${\hbox{\sf A}}{\hbox{\sf p}}_{ij}$ gives A with the $i$th and $j$th columns interchanged.

Interpreting the 1s in an $n\times n$ permutation matrix as Rooks gives an allowable configuration of nonattacking Rooks on an $n\times n$ Chessboard.

See also Elementary Matrix, Identity, Permutation, Rook Number