What this LU factorization calculator does
This matrix LU factorization calculator decomposes a square matrix into lower and upper triangular factors. In practical terms, it computes matrices L and U such that P·A = L·U when pivoting is enabled. Here, P is a permutation matrix that records row swaps needed for numerical stability.
LU decomposition is a core tool in linear algebra, numerical methods, engineering simulation, optimization, machine learning, and scientific computing. Once a matrix is decomposed, solving multiple systems with the same coefficient matrix becomes much faster.
How to use the calculator
Step-by-step
- Select the matrix size (for example, 3 for a 3×3 matrix).
- Click Generate Matrix to create editable inputs.
- Enter all matrix values (decimals and negatives are supported).
- Keep partial pivoting checked for better reliability.
- Click Calculate LU to see P, L, U, determinant, and verification error.
Understanding the output
Permutation matrix P
If pivoting is active, the algorithm may swap rows to avoid dividing by very small numbers. Those swaps are captured by matrix P.
Lower triangular matrix L
Matrix L has ones on its diagonal and elimination multipliers below the diagonal.
Upper triangular matrix U
Matrix U contains the transformed coefficients above and on the diagonal after elimination.
Why LU decomposition matters
Direct Gaussian elimination solves one right-hand side quickly, but LU decomposition is more reusable. If you need to solve A·x = b for many different vectors b, you compute LU once and use fast forward/back substitution for each new right-hand side.
- Efficient repeated solves in simulation and control systems
- Determinant computation from diagonal entries of U
- Foundation for advanced factorizations and preconditioning
Numerical stability and pivoting
Partial pivoting usually improves stability by selecting the largest absolute pivot in each column. Without pivoting, matrices with zero or near-zero pivots can fail even when a decomposition exists. For most real-world data, keep pivoting enabled.
Common pitfalls
- Entering a non-square matrix (LU here requires n×n input).
- Leaving blank cells in the matrix grid.
- Turning off pivoting for matrices that need row swaps.
- Assuming all singular matrices have LU decomposition with finite pivots.
Quick example matrix
Try this sample using the Load Example button:
[ [2, 1, 1], [4, -6, 0], [-2, 7, 2] ]
The calculator will produce valid L and U factors and show a small verification error close to zero, confirming the decomposition.