LU Decomposition Calculator
Decompose matrices into L and U factors effortlessly with our step-by-step calculator. Get instant solutions and detailed explanations for matrix factorization.
LU Decomposition Tool
Enter your square matrix and get the L and U factorization
Master LU Decomposition with Our Advanced Calculator
Our LU decomposition calculator is designed to help students, engineers, and researchers understand matrix factorization through the LU decomposition method. Whether you're working onlinear algebra homework, studying numerical analysis, or solvingsystems of linear equations, this tool provides comprehensive step-by-step solutions.
The LU decomposition calculator factors a square matrix into the product of a lower triangular matrix and an upper triangular matrix , such that . This matrix factorization calculator is particularly useful for numerical analysis and solving systems of linear equations efficiently. Our linear algebra calculator provides detailed steps showing how Gaussian elimination is used to obtain the decomposition.
Perfect for linear algebra students learning matrix factorization,engineering students working with numerical methods, and researchersanalyzing computational algorithms. The LU decomposition tool provides not just the L and U matrices, but also verification that their product equals the original matrix.
Types of Matrix Decompositions
| Decomposition Type | Formula | Applications | Complexity |
|---|---|---|---|
| LU Decomposition | Solving linear systems, matrix inversion | O(n³) | |
| QR Decomposition | Least squares, eigenvalue problems | O(n³) | |
| Cholesky | Positive definite matrices, optimization | O(n³) | |
| SVD | Data analysis, image processing | O(n³) | |
| Eigenvalue | Diagonalization, system analysis | O(n³) |
Common Mistakes to Avoid
Non-Square Matrices
LU decomposition is only defined for square matrices. For rectangular matrices, consider LUP decomposition or QR decomposition instead.
Zero Pivots
If a zero pivot is encountered during decomposition, the matrix may be singular. In such cases, consider using LUP decomposition with pivoting.
Uniqueness Issues
LU decomposition is not unique for singular matrices. Different algorithms may produce different L and U matrices for the same input.
How to Perform LU Decomposition
LU decomposition factors a square matrix A into the product of a lower triangular matrix L and an upper triangular matrix U. This is achieved through Gaussian elimination without row exchanges.
where is a lower triangular matrix with ones on the diagonal, and is an upper triangular matrix. The diagonal elements of are always 1, while the diagonal elements of are the pivots from Gaussian elimination.
Step-by-Step Method
Initialize matrices
Start with U = A and L = I (identity matrix)
Gaussian elimination
Perform elimination to get U, storing multipliers in L
Extract L matrix
The multipliers become the subdiagonal elements of L
Verify result
Check that A = LU by matrix multiplication
Key Properties
L Matrix Properties
Lower triangular with ones on diagonal: for all
U Matrix Properties
Upper triangular with pivots on diagonal: for
Examples
2×2 Matrix
A = [[2,1],[4,3]]
Process:
Verification:
3×3 Matrix
A = [[2,1,1],[4,-6,0],[-2,7,2]]
Process:
Perform Gaussian elimination
Store multipliers in L
Obtain upper triangular U
Verify decomposition
Singular Matrix
A = [[1,2],[2,4]]
Process:
Matrix is singular (det = 0)
LU decomposition may fail
Consider LUP decomposition
Use pivoting strategy
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Solve with AskMathAIFrequently Asked Questions
What is LU decomposition?
LU decomposition factors a square matrix A into the product of a lower triangular matrix L and an upper triangular matrix U, such that A = LU. L has ones on the diagonal, while U has the pivots from Gaussian elimination.
When is LU decomposition possible?
LU decomposition is possible for any square matrix that can be reduced to upper triangular form using Gaussian elimination without row exchanges. This includes most non-singular matrices.
What is the difference between LU and LUP decomposition?
LU decomposition uses Gaussian elimination without row exchanges, while LUP decomposition (also called PLU) includes row pivoting to handle cases where zero pivots would otherwise be encountered.
How is LU decomposition used to solve linear systems?
To solve Ax = b, first decompose A = LU, then solve Ly = b (forward substitution), and finally solve Ux = y (backward substitution). This is more efficient than direct Gaussian elimination.
Is LU decomposition unique?
For non-singular matrices, LU decomposition is unique if we require L to have ones on the diagonal. For singular matrices, the decomposition may not be unique or may not exist.
What are the computational advantages of LU decomposition?
Once A is decomposed into LU, solving multiple systems Ax = b for different b vectors becomes much faster, as you only need to perform forward and backward substitution instead of full Gaussian elimination.