1. What is LU Decomposition?

LU Decomposition is a method to factorize a matrix as the product of a lower triangular matrix (L) and an upper triangular matrix (U). This decomposition is used to solve systems of linear equations, invert matrices, and compute determinants more efficiently.

2. How Does the Calculator Work?

The calculator performs the following steps:

Where:

• \( A \) — Coefficient matrix (n×n)
• \( L \) — Lower triangular matrix with 1s on diagonal
• \( U \) — Upper triangular matrix
• \( b \) — Right-hand side vector
• \( x \) — Solution vector

3. Importance of LU Decomposition

Details: LU decomposition is computationally efficient for solving multiple systems with the same coefficient matrix but different right-hand sides. It's more stable than Gaussian elimination and forms the basis for many numerical algorithms.

4. Using the Calculator

Tips: Enter your square matrix A and vector b. The calculator supports matrices from 2×2 to 6×6. All values must be valid numbers.

5. Frequently Asked Questions (FAQ)

Q1: When does LU decomposition fail?
A: LU decomposition fails when the matrix is singular or when pivoting is required (though this implementation doesn't include pivoting).

Q2: How is this different from Gaussian elimination?
A: LU decomposition preserves the original matrix and factors it into L and U, allowing for efficient solving of multiple systems.

Q3: What's the computational complexity?
A: Decomposition is O(n³), while solving is O(n²) for each system.

Q4: Can I use this for non-square matrices?
A: No, LU decomposition requires square matrices.

Q5: What about numerical stability?
A: For better stability, consider adding partial pivoting (PA = LU).