1. What is PA LU Factorization?

Definition: PA LU factorization decomposes a square matrix into three matrices: a permutation matrix (P), a lower triangular matrix (L), and an upper triangular matrix (U).

Purpose: This decomposition is useful for solving systems of linear equations, calculating determinants, and finding matrix inverses more efficiently.

2. How Does PA LU Factorization Work?

The factorization follows the formula:

Where:

• \( A \) — Original square matrix
• \( P \) — Permutation matrix representing row exchanges
• \( L \) — Lower triangular matrix with 1s on the diagonal
• \( U \) — Upper triangular matrix

Explanation: The algorithm performs Gaussian elimination with partial pivoting to create the decomposition.

3. Importance of PA LU Factorization

Details: This factorization is numerically stable (due to partial pivoting) and forms the basis for many matrix computations in scientific computing and engineering.

4. Using the Calculator

Tips: Enter your square matrix size (2×2 to 6×6) and fill in all matrix elements. The calculator will compute the P, L, and U matrices.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between LU and PA LU?
A: PA LU includes partial pivoting (row exchanges) for numerical stability, represented by the P matrix.

Q2: When does PA LU factorization fail?
A: It fails when the matrix is singular (has zero determinant), though partial pivoting makes it more robust than basic LU.

Q3: How is the permutation matrix P determined?
A: P records all row exchanges made during the elimination process to avoid division by small numbers.

Q4: What are practical applications of PA LU?
A: Solving linear systems, computing determinants, matrix inversion, and as a step in eigenvalue algorithms.

Q5: Why are L's diagonal elements 1?
A: This is a convention that makes the factorization unique (when combined with partial pivoting).