LU Decomposition Calculator (with Partial Pivoting)
Enter a square matrix A. This tool computes P, L, U such that P·A = L·U.
What is LU decomposition?
LU decomposition is a core technique in linear algebra where a square matrix is factored into two triangular matrices. In practice, we usually compute a permutation matrix too, so the full relationship is: P·A = L·U.
- P is a permutation matrix (tracks row swaps for numerical stability).
- L is lower triangular with ones on the diagonal.
- U is upper triangular.
Why this matrix LU decomposition calculator is useful
LU factorization is not just a classroom exercise. It is a practical computational tool used in engineering, physics, machine learning, and numerical simulation. Once you decompose one matrix, you can solve multiple systems quickly.
Common use cases
- Solving linear systems of equations: A·x = b
- Computing matrix determinants efficiently
- Building inverse matrices (when the matrix is non-singular)
- Speeding up repeated computations in scientific code
How to use the calculator
- Choose matrix size n (between 2 and 8).
- Click Generate Matrix Grid.
- Enter each matrix value.
- Click Compute LU Decomposition.
The result panel displays P, L, and U, plus a numerical check using max |P·A - L·U|. A small value means the factorization is correct up to floating-point rounding.
Algorithm notes
This page uses the Doolittle-style LU approach with partial pivoting. Partial pivoting chooses the largest available pivot in each column and swaps rows before elimination. This improves numerical stability and avoids division by zero in many practical situations.
What happens if the matrix is singular?
If a pivot is zero (or extremely close to zero), the matrix is singular or nearly singular for this process. In that case, the calculator reports an error and no valid LU decomposition is returned.
Determinant from LU decomposition
After factorization, the determinant can be computed fast: det(A) = (-1)s × product of diagonal entries of U, where s is the number of row swaps.
Practical tips
- Use pivoting for better stability, especially with real-world data.
- Very large or tiny numbers can create floating-point round-off effects.
- If your matrix is close to singular, expect sensitivity in results.
- For large-scale work, use optimized libraries (LAPACK, NumPy, MATLAB, Julia LinearAlgebra).
Summary
A matrix LU decomposition calculator helps you quickly factor matrices, verify decomposition structure, and understand how elimination works under the hood. Whether you are studying linear algebra or building a numerical system solver, LU decomposition is one of the most important tools in your toolkit.