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).

Set matrix dimension

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

Pivoting method:
Display values as:
1 — Set Up
Loads a pre-filled 3×3 matrix — ready to factor.
2 — Compute
Performs LU decomposition with partial pivoting, showing P, L, and U step by step.
Resets all dimensions, matrix entries, and results.