PA = LU Factorization

Compute the PA = LU factorization¹ of a square matrix with various pivoting strategies.
Essential for solving linear systems², computing determinants, and matrix inversion.

For a square matrix \(A\), the LU factorization with pivoting is:

\[ PA = LU \]

Components:

\(P\) is a permutation matrix³ (row exchanges)
\(L\) is a unit lower triangular matrix (diagonal entries are 1)
\(U\) is an upper triangular matrix

Pivoting Strategies:

Partial Pivoting: Choose the largest magnitude element in the current column
Scaled Partial Pivoting: Account for row scaling to reduce errors⁴
No Pivoting: No row exchanges (may fail for some matrices)

Used to solve \(Ax = b\) by solving \(Ly = Pb\) then \(Ux = y\) (both triangular systems).

Matrix size \(n\):

Enter the matrix size n and click Generate Matrix to create an n×n input grid.

Pivoting method: Partial Pivoting Scaled Partial Pivoting No Pivoting

Display values as: Decimal Fraction

1 — Compute

Performs LU decomposition with partial pivoting, showing P, L, and U step by step.

Resets all dimensions, matrix entries, and results.

Footnotes

The PA = LU factorization decomposes a square matrix \(A\) into the product of a permutation matrix \(P\), a unit lower triangular matrix \(L\), and an upper triangular matrix \(U\), such that \(PA = LU\). It is a fundamental tool in numerical linear algebra. ↑
To solve \(Ax = b\), apply the factorization: first solve \(Ly = Pb\) by forward substitution, then solve \(Ux = y\) by back substitution. Both are \(O(n^2)\) operations once the \(O(n^3)\) factorization is complete. ↑
A permutation matrix \(P\) is an orthogonal matrix obtained by permuting the rows of the identity matrix. It satisfies \(P^{-1} = P^T\) and \(\det(P) = \pm 1\). ↑
Scaled partial pivoting selects the pivot row by comparing \(|a_{ik}| / s_i\), where \(s_i = \max_j |a_{ij}|\) is the scale factor for row \(i\). This strategy reduces the effect of poorly scaled rows on numerical stability. ↑

Cite this tool

DOI: https://doi.org/10.5281/zenodo.20981296

Kapita, S. (2026). PA = LU Factorization. Math Tools. https://doi.org/10.5281/zenodo.20981296

Kapita, Shelvean. "PA = LU Factorization." Math Tools, 2026, doi.org/10.5281/zenodo.20981296.

@online{kapita2026lufactor,
  author       = {Shelvean Kapita},
  title        = {{PA = LU Factorization}},
  year         = {2026},
  organization = {Math Tools},
  doi          = {10.5281/zenodo.20981296},
  url          = {https://doi.org/10.5281/zenodo.20981296}
}