LU Decomposition Calculator

LU decomposition factors a square matrix A into the product of a lower-triangular matrix L and an upper-triangular matrix U: A = LU. This is one of the most useful matrix factorizations because it turns the problem of solving Ax = b into two easy triangular solves: Ly = b (forward substitution), then Ux = y (back substitution).

This calculator uses Doolittle’s algorithm, which forces L to have 1s on its diagonal. The diagonal of U then absorbs the scaling. The alternative convention (Crout) puts 1s on U’s diagonal — the two are equivalent up to a diagonal scaling.

Once you have L and U, solving multiple systems with the same A but different right-hand sides is essentially free: the expensive O(n³) factorization happens only once. This is why direct linear solvers in numerical libraries factor the matrix first and reuse it.

LU exists for any square matrix where you don’t hit a zero pivot during elimination. Some matrices need row permutations first (PA = LU, called PLU decomposition with a permutation matrix P) to avoid dividing by zero. This calculator does not pivot — if a zero appears on the diagonal during elimination, it returns an error and you should rearrange the input rows manually.

Worked example: A = [[4,3],[6,3]]. Doolittle gives L = [[1,0],[1.5,1]], U = [[4,3],[0,-1.5]]. Check: L*U = [[4,3],[6,3]]. ✓

The determinant of A equals the product of U’s diagonal entries (since L’s diagonal is all 1s and det(LU) = det(L)det(U)). For the example: det = 4 * (-1.5) = -6. Matches the direct 2x2 computation 43 - 3*6 = -6. ✓

LU is preferred over computing A inverse explicitly. Inverting is wasteful — you almost never need the inverse itself, only its action on a vector, which the LU factorization gives you faster and more accurately.