For a square matrix $A$, the LU factorization with pivoting is:

$$ PA = LU $$

Components:

• $P$ is a permutation matrix (row exchanges)
• $L$ is a unit lower triangular matrix (diagonal entries are 1)
• $U$ is an upper triangular matrix

Pivoting Strategies:

• Partial Pivoting: Choose largest magnitude element in current column
• Scaled Partial Pivoting: Account for row scaling to reduce errors
• No Pivoting: No row exchanges (may fail for some matrices)

Used to solve $Ax = b$ by solving $Ly = Pb$ then $Ux = y$ (both triangular systems).