Row pivoting

• row swapping can help us avoid division by zero
• pivot elements: diagonal elements of the upper triangular matrix $\mathbf{U}$
• row pivoting: when doing LU-factorization in column $j$, choose the row index (aka pivot element) that corresponds to the largest absolute value to swap rows
• Theorem: row-pivoted LU factorization works iff original matrix is invertible (i.e., is non-singular)