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LU Decomposition

LU decomposition calculator. Enter A to compute L, U and a permutation P with partial pivoting—for matrix factorization.

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P A = L U
space-separated entries, semicolon-separated rows, e.g. 1 2;3 4

📖 Tutorial｜LU Decomposition

1. Definition
Factor A into a lower triangular L and upper triangular U: PA=LU (P is a permutation).
2. Symbols
SymbolMeaning
Lunit lower triangular
Uupper triangular
Ppermutation matrix
3. How this tool works
  • LU is the matrix form of Gaussian elimination;
  • L stores multipliers, U is the upper triangular result;
  • Partial pivoting picks the largest entry for stability.
4. Steps
  1. Enter A;
  2. Click Calculate to see L, U and row swaps.
5. Example
Example: A=2 −1 0;−1 2 −1;0 −1 2. L is unit lower triangular, U is upper triangular, and L·U=A.
6. Pitfalls
Swap rows when a pivot is 0;
Diagonal entries of L are 1;
LU is used to solve many Ax=b quickly.

❓ FAQ｜LU Decomposition

What is LU decomposition?
Factor A into lower triangular L and upper triangular U.
Why P?
Partial pivoting swaps rows; P records the permutation.
Why are L's diagonal entries 1?
Doolittle convention: L is unit lower triangular.
Why use LU?
Factor once, then solve many different b by forward/back substitution.
Pivot is 0?
Swap rows; if still 0, A is singular.