LU Decomposition Calculator (with Partial Pivoting)
Enter a square matrix and compute P, L, U such that P·A = L·U. This tool supports matrix sizes from 2×2 to 8×8.
Tip: decimals and negative values are allowed.
What Is LU Decomposition?
LU decomposition factors a square matrix A into two triangular matrices: a lower triangular matrix L and an upper triangular matrix U. In practical numerical computing, we often include a permutation matrix P to improve stability, giving:
P·A = L·U
This factorization is a core building block in linear algebra and is widely used for solving systems of equations, matrix inversion, and determinant computation.
How to Use This Calculator
- Choose matrix size (2 to 8).
- Click Generate Matrix Inputs.
- Fill in every matrix entry.
- Click Compute LU Decomposition.
- Review P, L, U, determinant, and residual error.
Understanding the Output
L Matrix (Lower Triangular)
The L matrix contains multipliers used during Gaussian elimination. It has 1s on the main diagonal and zeros above the diagonal.
U Matrix (Upper Triangular)
The U matrix is the transformed version of the original matrix after elimination. All values below the diagonal are zero.
P Matrix (Permutation Matrix)
The P matrix records row swaps from partial pivoting. Pivoting makes decomposition more stable and avoids dividing by very small numbers when possible.
Why LU Decomposition Is Useful
- Fast repeated solves: If you need to solve A·x = b for many different b vectors, decompose once and reuse.
- Determinant: For PA = LU, the determinant is the product of diagonal entries of U adjusted by row-swap sign.
- Matrix inverse: Inversion can be performed through repeated triangular solves after decomposition.
- Numerical methods: LU appears in optimization, simulation, machine learning, and scientific computing.
Common Pitfalls
Singular Matrices
If a matrix is singular (determinant near zero), LU decomposition may fail or become unstable. The calculator will report an error in this case.
Rounding Effects
Floating-point arithmetic introduces small rounding errors. That is why this tool also reports a residual metric: the maximum absolute value in P·A − L·U.
Input Completeness
Every matrix entry must be numeric. Blank cells or invalid values will trigger validation messages.
Quick Example
Click Load Example to populate a sample 3×3 matrix that requires pivoting. This lets you inspect how row swaps alter P while still yielding a valid decomposition.
Final Thoughts
LU decomposition is one of the most practical matrix factorizations in applied mathematics. Whether you are studying numerical linear algebra or building engineering software, understanding LU and pivoting is a major step toward reliable computation.