Columnwise partial pivoting ($PA=LU$)

At each step of Gaussian elimination, select the largest absolute entry in the uneliminated part of the current column and interchange rows to put it on the diagonal. A nonsingular trailing Schur complement has a nonzero entry in its first column, so this procedure always finds a nonzero pivot in exact arithmetic. The permutation matrix $P$ records row interchanges; earlier multipliers must be interchanged too when assembling the lower-triangular factor. This yields the LU decomposition $PA=LU$ with unit diagonal in $L$. Searching within columns and interchanging rows is distinct from interchanging columns.