1. What is LU Decomposition?

LU decomposition factors a matrix as the product of a lower triangular matrix (L) and an upper triangular matrix (U). The L matrix has 1s on its diagonal, while U has non-zero values on its diagonal.

2. How Does the Calculator Work?

The calculator performs LU decomposition using Doolittle's algorithm:

Where:

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

Explanation: The algorithm systematically decomposes the matrix by solving for elements of L and U in a specific order.

3. Importance of LU Decomposition

Details: LU decomposition is fundamental for solving systems of linear equations, computing determinants, and matrix inversion. It's more efficient than Gaussian elimination when solving multiple equations with the same coefficient matrix.

4. Using the Calculator

Tips: Enter your square matrix with rows separated by semicolons and columns separated by commas. The matrix must be square (n×n) and non-singular.

5. Frequently Asked Questions (FAQ)

Q1: What matrices can be LU decomposed?
A: Any square matrix that can be reduced to row echelon form without row exchanges (i.e., all leading principal minors are non-zero).

Q2: How is LU different from QR decomposition?
A: LU decomposes into triangular matrices while QR decomposes into an orthogonal matrix and upper triangular matrix.

Q3: What are applications of LU decomposition?
A: Solving linear systems, computing determinants, matrix inversion, and numerical analysis applications.

Q4: What if my matrix requires pivoting?
A: This calculator implements basic LU decomposition. For matrices requiring pivoting, PA = LU decomposition would be needed.

Q5: Can I decompose non-square matrices?
A: No, LU decomposition requires square matrices. For rectangular matrices, consider QR or SVD decomposition.