Choose a size between 2 and 8, then generate the input grid.
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 work, row swaps are often needed, so the most reliable form is:
P × A = L × U
where P is a permutation matrix that captures row exchanges used during partial pivoting.
Why use an LU matrix calculator?
LU decomposition is one of the most useful tools in linear algebra and scientific computing. It helps you:
- Solve linear systems efficiently, especially with multiple right-hand sides.
- Compute determinants quickly from the diagonal entries of U (plus row-swap sign).
- Build matrix inverses in a structured way.
- Understand numerical stability through pivoting behavior.
How this calculator works
1) Build your matrix
Select a size, generate the grid, and fill the entries. You can use decimals or negatives.
2) Partial pivoting
The calculator uses partial pivoting: at each elimination step, it selects the row with the largest absolute pivot in the current column. This reduces round-off problems and avoids dividing by tiny numbers.
3) Outputs you get
- P (permutation matrix)
- L (lower triangular matrix with 1s on the diagonal)
- U (upper triangular matrix)
- Determinant estimate and residual check for accuracy
Interpreting your results
A small residual (close to zero) means the decomposition is numerically consistent: max |(P×A) - (L×U)| should be tiny.
If the calculator reports a singular or near-singular matrix, the decomposition may fail because at least one pivot is essentially zero.
Common use cases
- Engineering: circuit equations, structural analysis, and simulation models.
- Data science: repeated linear solves inside optimization loops.
- Education: learning Gaussian elimination and matrix factorization.
- Research: determinant and conditioning diagnostics.
Tips for best accuracy
- Prefer scaled, well-conditioned matrices when possible.
- Use pivoting (already enabled here) for stable decomposition.
- Watch out for extremely large or tiny coefficients in the same matrix.
- Validate with the residual shown in the output.
Quick FAQ
Does every matrix have LU decomposition without pivoting?
No. Some require row swaps, which is exactly why the calculator returns P as well.
Can this solve Ax = b directly?
This tool focuses on factorization. Once you have L and U, you can solve using forward and backward substitution.
What if determinant is zero?
Then the matrix is singular (non-invertible), and some systems may have no unique solution.