What is LU factorization?

LU factorization is a matrix factorization method that results in a factor that is the product of a lower triangular matrix and an upper triangular matrix through the use of a sequence of Gaussian elimination steps. This product may also include a permutation matrix.

LU factorization is used to factor square matrices through the use of permutations to obtain a lower triangular matrix L and an upper triangular matrix U as the final factors. This process is represented as A = LU or A = PLU, where P is a permutation matrix. A procedural problem may arise if the leading diagonal element becomes zero.