LU Factorization Calculator
Input Matrix A (3×3)
Enter each row on a new line, with values separated by commas
Understanding LU Factorization
LU Factorization (or LU Decomposition) expresses a matrix A as the product of a lower triangular matrix L and an upper triangular matrix U.
A = LU or PA = LU (with pivoting)
Matrix L
- Lower triangular (zeros above diagonal)
- Ones on the diagonal
- Stores elimination multipliers
Matrix U
- Upper triangular (zeros below diagonal)
- Result of Gaussian elimination
- Diagonal gives determinant info
Why Use Pivoting?
Partial pivoting exchanges rows to ensure the pivot element (diagonal entry used for elimination) has the largest absolute value in its column.
- Prevents division by zero
- Improves numerical stability
- Reduces round-off errors
- Required when diagonal elements are zero
Applications
Solving Ax = b
Solve Ly = b (forward substitution), then Ux = y (back substitution)
Determinant
det(A) = det(L) × det(U) = product of U's diagonal
Matrix Inverse
Solve AX = I column by column using LU
Algorithm Complexity
LU Factorization has time complexity O(n³) for an n×n matrix. However, once computed, solving Ax = b for new right-hand sides b only requires O(n²) operations (forward and back substitution).
About LU Factorization Calculator - Matrix Decomposition
Compute LU factorization of matrices with partial pivoting. Decompose into lower and upper triangular matrices with step-by-step solutions.
Our **LU Factorization Calculator** decomposes square matrices into the product of lower (L) and upper (U) triangular matrices. Uses Gaussian elimination with optional partial pivoting (PA = LU). Essential for solving linear systems efficiently.
LU factorization is the matrix version of Gaussian elimination stored in reusable form. Once computed, you can solve Ax = b for multiple right-hand sides b efficiently using forward and back substitution.
The calculator shows each elimination step, multiplier computation, and row operation. Understand how L stores the elimination multipliers while U becomes the row-echelon form. Compute determinants as the product of U's diagonal.
Key Features
Why Use This Tool?
Common Use Cases
Linear Systems: Solve Ax = b efficiently.
Determinants: det(A) = product of U diagonal.
Matrix Inverse: Solve AX = I column by column.
Numerical Analysis: Stable matrix computations.
Related Tools
How to Use
Enter matrix size
Input matrix elements
Enable/disable partial pivoting
Click Compute
View L, U, and P matrices