How this calculator works
LU decomposition records the row operations of Gaussian elimination. L holds the multipliers below the diagonal, and U is the resulting upper triangular matrix. With partial pivoting, rows are swapped first and the swaps are collected in P, giving P A = L U. The multipliers are stored in the permuted order, so L remains unit lower triangular. If a pivot column is entirely zero, the matrix is singular but the factors still exist.
Formula
P A = L U, where L is unit lower triangular and U is upper triangular.
Worked example
For the matrix with rows 4, 3 and 2, 1, no row swap is needed. Eliminating the 2 with a multiplier of 0.5 gives L as 1, 0; 0.5, 1 and U as 4, 3; 0, -0.5. Their product is 4, 3; 2, 1. The matrix with rows 0, 2 and 1, 3 needs a swap, so P is 0, 1; 1, 0.
Advertisement
Frequently asked questions
Why is there a permutation matrix?
Partial pivoting swaps rows to keep the largest available pivot. The swaps are recorded in P so the identity P A = L U remains exact.
When is LU decomposition unique?
Without row swaps, a unit lower triangular L and upper triangular U are unique if every leading principal minor is non-zero. Pivoting can make the factors differ while still satisfying P A = L U.
Can a singular matrix have an LU factorization?
Yes. A singular matrix can still be factored, usually with a zero on the diagonal of U. The factor does not give an inverse, but it still records the elimination steps.