🔲 LU Decomposition Calculator
Factor matrix into Lower and Upper triangular matrices
How to Use This Calculator
Select Matrix Size
Choose the size of your square matrix (2×2, 3×3, or 4×4).
Enter Matrix Elements
Input all elements of your square matrix.
Calculate
Click to decompose matrix into L (lower triangular) and U (upper triangular) matrices.
Verify
Check that L × U equals the original matrix A.
Formula
A = L × U
Lower triangular × Upper triangular
Definition:
LU decomposition factors a square matrix A into the product of a lower triangular matrix L (with 1s on diagonal) and an upper triangular matrix U.
Structure:
- L has 1s on diagonal, zeros above diagonal
- U has zeros below diagonal
- Typically uses Crout's or Doolittle's algorithm
Requirements:
- Matrix must be square
- All leading principal minors must be nonzero
- Not all matrices have LU decomposition without pivoting
About LU Decomposition Calculator
The LU Decomposition Calculator factors a square matrix into the product of lower and upper triangular matrices. This decomposition is fundamental for solving linear systems, computing determinants, and matrix inverses.
When to Use This Calculator
- Linear Systems: Solve Ax = b efficiently
- Determinants: Compute det(A) = det(L) × det(U)
- Matrix Inverses: Find A⁻¹ via triangular solves
- Numerical Methods: Improve computational efficiency
- Eigenvalues: Preparation for eigenvalue algorithms
Why Use Our Calculator?
- ✅ Crout's Method: Efficient LU decomposition algorithm
- ✅ Verification: Shows L × U = A
- ✅ Clear Display: Visualizes L and U matrices
- ✅ Educational: Helps understand matrix factorization
- ✅ Accurate: Precise calculations
- ✅ Free: No registration required
Key Concepts
- Lower Triangular: Zeros above diagonal, 1s on diagonal (for L)
- Upper Triangular: Zeros below diagonal
- Decomposition: Factorization A = L × U
- Pivoting: Sometimes needed for numerical stability (PA = LU)
- Uniqueness: With constraints, LU decomposition is unique
Applications
Solving Systems: Factor once, solve multiple times with different b vectors.
Determinant: det(A) = det(L) × det(U) = (product of L diagonal) × (product of U diagonal).
Frequently Asked Questions
Does every matrix have LU decomposition?
No. A matrix has LU decomposition if all leading principal minors are nonzero. Some matrices require pivoting (PA = LU) or may need different decompositions.
Why are the diagonals of L all 1?
This is Crout's version of LU decomposition. L has 1s on diagonal. Alternatively, Doolittle's method puts 1s on U's diagonal and uses a different L.
Can I decompose non-square matrices?
Traditional LU requires square matrices. For rectangular matrices, use LUP or QR decomposition.
What if the decomposition fails?
If a pivot is zero, the matrix may need pivoting or doesn't have standard LU form. Try rearranging rows or use other decompositions.
How is this different from Cholesky decomposition?
Cholesky is for symmetric positive-definite matrices: A = LLᵀ. LU is for general square matrices: A = LU. Cholesky is faster when applicable.