LU Factorization Calculator

What Is LU Factorization?

LU factorization (or LU decomposition) breaks a matrix A into the product of a lower triangular matrix L and an upper triangular matrix U, such that A = LU. This is one of the most important algorithms in linear algebra and numerical computing.

LU Decomposition:
A = L × U

L = Lower triangular (1s on diagonal)
U = Upper triangular

Used to solve Ax = b by solving Ly = b then Ux = y

The Algorithm

LU factorization uses Gaussian elimination. As you eliminate entries below the diagonal, the multipliers form the L matrix, and the resulting upper triangular matrix is U. For a 2×2 matrix: compute U through row operations, and L records the multipliers.

Applications

Solving systems of linear equations efficiently
Computing matrix determinants (det(A) = det(L)×det(U))
Computing matrix inverses
Numerical simulations and engineering computations

How to Use

Enter the elements of a 2×2 matrix. The calculator performs LU factorization and shows both the L and U matrices.

Partial Pivoting

In practice, LU factorization often uses partial pivoting (row swaps) for numerical stability. This produces PA = LU, where P is a permutation matrix.

When Does LU Exist?

LU factorization exists for any matrix where all leading principal minors are non-zero. If a zero pivot is encountered, pivoting is required.

Frequently Asked Questions

LU factorization decomposes a matrix A into a lower triangular matrix L and upper triangular matrix U such that A = LU.

It allows efficient solving of multiple linear systems with the same coefficient matrix but different right-hand sides.

You need partial pivoting (row swaps) to handle zero pivots, giving PA = LU.

det(A) = det(L) × det(U) = product of diagonal entries of U (since L has 1s on diagonal).

Not directly. Matrices with zero leading principal minors require pivoting. With pivoting, any non-singular matrix can be decomposed.

LU uses Gaussian elimination; QR uses orthogonal transformations. QR is more numerically stable but slower.