 LU decomposition

An LU decomposition writes a square matrix as $A=LU$, with $L$ a lower triangular matrix and $U$ an upper triangular matrix. We use a unit triangular matrix $L$, meaning $l_{ii}=1$. The factors record Gaussian elimination: $U$ contains the reduced equations, while $L$ stores the elimination multipliers needed to reconstruct the original equations. This makes Reusing LU factors for multiple right-hand sides inexpensive.^[ref]

For an invertible matrix, unpivoted LU decomposition requires nonzero pivot elements at every elimination step; invertibility alone does not guarantee this. In general use Pivoted LU decomposition, with the explicit convention $PA=LU$ and a permutation matrix $P$. Once the factors exist, solving a linear system $Ax=b$ needs forward substitution followed by back substitution, rather than forming a matrix inverse.

LU decomposition is a matrix factorization technique used in numerical linear algebra. It involves breaking down a square matrix \( A \) into the product of two matrices: a lower triangular matrix \( L \) and an upper triangular matrix \( U \).