What Is an LU Factorization Calculator?
An LU factorization calculator decomposes a square matrix A into the product of a lower triangular matrix L and an upper triangular matrix U, so that A = LU. This decomposition is a cornerstone of numerical linear algebra, used to solve systems of equations, compute determinants, and invert matrices efficiently. Enter your matrix and the calculator returns the L and U factors.
How to Use the Calculator
- Enter your square matrix.
- Calculate — see the lower (L) and upper (U) triangular matrices.
What LU Decomposition Is
LU factorization writes A = LU, where L has ones on the diagonal and entries below it, and U has the entries on and above the diagonal. It is essentially Gaussian elimination recorded in matrix form, with U being the row-echelon result and L storing the elimination multipliers.
Why LU Factorization Is Useful
| Task | How LU Helps |
|---|---|
| Solving Ax = b | Solve Ly = b then Ux = y by simple substitution |
| Multiple right-hand sides | Reuse L and U without redoing elimination |
| Determinant | Product of U's diagonal entries |
| Matrix inverse | Solve for each column efficiently |
Solving Systems with LU
Once A = LU, solving Ax = b becomes two easy steps: first solve Ly = b by forward substitution, then solve Ux = y by back substitution. This is much faster than re-running elimination, especially when solving for many different right-hand sides.
Pivoting
Sometimes rows must be swapped for numerical stability or to avoid zero pivots, giving PA = LU, where P is a permutation matrix. This variant is the standard form used in practice.
Frequently Asked Questions
What is LU factorization?
LU factorization decomposes a matrix A into a lower triangular matrix L and an upper triangular matrix U such that A = LU, derived from Gaussian elimination.
Why use LU decomposition?
It makes solving linear systems efficient, especially with multiple right-hand sides, and provides a fast way to compute determinants and inverses.
How do you solve a system with LU?
Solve Ly = b by forward substitution, then Ux = y by back substitution. Both steps are quick because L and U are triangular.
What is partial pivoting?
Partial pivoting swaps rows to avoid zero or small pivots for numerical stability, giving PA = LU with a permutation matrix P.
Is this LU factorization calculator free?
Yes — it is completely free, requires no signup, and returns the L and U matrices.