LU Decomposition Calculator – Factor Matrices into L and U
Decompose matrices into lower and upper triangular factors
How to Use
- Select matrix size (2x2 or 3x3)
- Enter the matrix elements
- Click calculate to see the LU decomposition
- Review the L and U matrices with verification
What is LU Decomposition?
LU decomposition (also called LU factorization) is a method of factoring a matrix A into the product of a lower triangular matrix L and an upper triangular matrix U, such that A = LU. The lower triangular matrix has all zeros above the diagonal, while the upper triangular matrix has all zeros below the diagonal.
This decomposition is fundamental in numerical linear algebra and is used extensively for solving systems of linear equations, computing determinants, and finding matrix inverses efficiently.
Applications of LU Decomposition
- Solving systems of linear equations (Ax = b)
- Computing matrix determinants efficiently
- Finding matrix inverses
- Numerical stability in computational algorithms
- Engineering and physics simulations
Doolittle's Method
This calculator uses Doolittle's algorithm, which sets the diagonal elements of L to 1. The algorithm systematically computes the elements of U row by row and L column by column, ensuring that A = LU.
Frequently Asked Questions
- What does it mean if LU decomposition fails?
- LU decomposition without pivoting fails when a zero pivot is encountered, meaning the matrix is singular or the decomposition requires row exchanges. In such cases, LU decomposition with partial pivoting (PA = LU) should be used.
- Why is LU decomposition useful?
- Once a matrix is decomposed into L and U, solving systems of equations becomes much faster. Instead of solving Ax = b directly, you solve Ly = b (forward substitution) and then Ux = y (back substitution), which is computationally efficient.
- What is the difference between L and U matrices?
- L (Lower triangular) has non-zero elements only on and below the main diagonal, with 1s on the diagonal in Doolittle's method. U (Upper triangular) has non-zero elements only on and above the main diagonal.