Definition of permutation matrix

Definition: A permutation matrix is a square binary matrix obtained by permuting the rows (or columns) of the identity matrix. Each row and each column contains exactly one 1, with all other entries being 0.

Example: Permuting rows of $I_3$:
$$P = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}$$

Properties:

• $P^{-1} = P^T$ (permutation matrices are orthogonal)
• $\det(P) = \pm 1$ (sign depends on permutation parity)
• Multiplying $PA$ permutes the rows of $A$
• Multiplying $AP^T$ permutes the columns of $A$
• $P^k = I$ for some positive integer $k$

Permutation matrices are used in LU decomposition with partial pivoting ($PA = LU$) and in representing permutations as linear transformations.