Overview
LU decomposition factors a matrix A into the product of a lower triangular matrix L and an upper triangular matrix U. This makes solving systems Ax = b efficient: first solve Ly = b (forward substitution), then Ux = y (back substitution).
Instead of solving the full system directly, you 'pre-process' the matrix into two simpler triangular forms. This is especially useful when you need to solve the same system with multiple right-hand sides — the LU factorization only needs to be done once.
LU Decomposition
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Mathematical Foundation
Formal Statement
Required Conditions
- • The matrix A must be square
- • All leading principal minors of A must be non-zero (for LU without pivoting)
- • L has 1s on its diagonal; U has the pivots on its diagonal
Common Mistakes & Pitfalls
- ⚠ Confusing which matrix (L or U) has 1s on the diagonal — L has 1s by convention.
- ⚠ Not recognizing when LU decomposition without pivoting fails (zero pivot).
- ⚠ Mixing up forward and back substitution — Ly = b first, then Ux = y.