= Columnwise partial pivoting
{title2=$PA=LU$}
= Partial pivoting
{synonym}
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.
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