1. What is LU Factorization?

Definition: LU factorization decomposes a square matrix into the product of a lower triangular matrix (L) and an upper triangular matrix (U).

Purpose: This decomposition is useful for solving systems of linear equations, matrix inversion, and calculating determinants.

2. How Does LU Factorization Work?

The decomposition follows the formula:

Where:

• \( A \) — Original square matrix
• \( L \) — Lower triangular matrix (with 1s on diagonal)
• \( U \) — Upper triangular matrix

Explanation: The algorithm systematically eliminates matrix elements to create the triangular forms while preserving the original matrix's information.

3. Importance of LU Factorization

Details: LU decomposition is more efficient than Gaussian elimination when solving multiple equations with the same coefficient matrix but different right-hand sides.

4. Using the Calculator

Tips: Enter your square matrix elements (up to 6×6). The calculator will display the L and U matrices with 4 decimal precision.

5. Frequently Asked Questions (FAQ)

Q1: Does every matrix have an LU decomposition?
A: No, the matrix must be square and its leading principal minors must be non-singular. Some matrices require permutation (PLU decomposition).

Q2: What's the difference between LU and QR decomposition?
A: LU uses triangular matrices while QR decomposes into an orthogonal matrix (Q) and upper triangular matrix (R).

Q3: How is LU decomposition used in solving equations?
A: After decomposition, solve Ly = b then Ux = y, which is computationally efficient.

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

Q5: Why might the decomposition fail?
A: If a zero pivot is encountered during decomposition, row exchanges (permutation) are needed.