permutation matrix

The identity matrix with its rows reordered; used to record row swaps.

A permutation matrix $P$ is the identity matrix with its rows reordered — equivalently, a square matrix with exactly one $1$ in every row and column. Left-multiplying $PA$ reorders $A$ 's rows the same way. Every permutation matrix is invertible with $P^{-1}=P^{\mathsf T}$ . Recording every swap elimination needs up front gives the fully general factorization $PA=LU$ .