Online LU Matrix Decomposition Calculator
What is LU Decomposition and Why is it Important?
LU decomposition (also called LU factorization) is a fundamental technique in linear algebra that decomposes a square matrix \(A\) into the product of a lower triangular matrix \(L\) and an upper triangular matrix \(U\):
\[ A = L \times U \]
Importance of LU Decomposition:
- Solving Systems of Linear Equations: Once we have \(A = LU\), we can solve \(Ax = b\) by first solving \(Ly = b\) (forward substitution) and then \(Ux = y\) (backward substitution), which is much faster than direct methods.
- Computing Determinants: The determinant of \(A\) equals the product of the diagonal elements of \(U\) (if \(L\) has ones on its diagonal), making determinant calculation efficient.
- Matrix Inversion: LU decomposition provides an efficient method for finding the inverse of a matrix.
- Numerical Stability: When combined with partial pivoting (LUP decomposition), it provides stable solutions to linear systems.
- Computational Efficiency: It reduces the computational complexity from \(O(n^3)\) to approximately \(O(n^2)\) for solving multiple systems with the same coefficient matrix.
How LU Decomposition is Calculated:
The decomposition is typically computed using Gaussian elimination. Here are the key steps:
- Step 1: Start with the original matrix \(A\).
- Step 2: For each column \(k\) from 1 to \(n-1\):
- For each row \(i\) below the diagonal (\(i > k\)):
- Compute the multiplier: \(m_{ik} = \frac{a_{ik}}{a_{kk}}\)
- Store \(m_{ik}\) in the \(L\) matrix at position \((i,k)\)
- Subtract \(m_{ik}\) times row \(k\) from row \(i\) to eliminate elements below the diagonal in column \(k\)
- Step 3: The resulting matrix after all eliminations becomes the upper triangular matrix \(U\).
- Step 4: Construct \(L\) as a lower triangular matrix with ones on the diagonal and the multipliers \(m_{ik}\) in the appropriate positions.
Note: Not all matrices have an LU decomposition. A matrix has an LU decomposition if and only if all its leading principal minors (determinants of the top-left \(k \times k\) submatrices) are non-zero.
LU Matrix Decomposition
An online calculator that calculates the LU decomposition of a square matrix.
An \( LU \) decomposition of a given matrix \( A \) can be written as:
\[ A = L U \]
where \( L \) is a lower triangular matrix and \( U \) is an upper triangular matrix.
Note: Not all square matrices have an LU decomposition. An invertible matrix \( A \) has an \( LU \) decomposition if and only if all its leading principal minors are different from zero.
Note: Any decomposition with outputs such as "NaN" or "Infinity" is not valid and means that the matrix does not have an LU decomposition.
More References and Links
- LU Decomposition
- Row Reduce Augmented Matrices - Calculator
- Linear Algebra Calculators
- Linear Algebra - Questions with Solutions
- Linear Algebra and its Applications - 5th Edition - David C. Lay, Steven R. Lay, Judi J. McDonald
- Elementary Linear Algebra - 7th Edition - Howard Anton and Chris Rorres