Linear Algebra — 5.1: Permutation Matrices and PA = LU

Описание: When a zero appears in the pivot position, standard LU factorization fails — but the matrix itself may be perfectly invertible. This video shows how permutation matrices fix the problem by recording row swaps, upgrading A = LU to the universal factorization PA = LU that works for every invertible matrix.

Key concepts covered:
• Why a zero pivot breaks standard elimination and how row exchanges resolve it
• LU factorization review: lower triangular L (with multipliers) times upper triangular U
• Permutation matrices as reordered identity matrices, with one entry of 1 per row and column
• Counting permutation matrices: n! possible n×n permutations
• The key property P⁻¹ = Pᵀ, verified through explicit multiplication
• Full worked example: factoring A = [[0,1,1],[1,2,1],[2,7,9]] into PA = LU step by step
• A = LU as the special case where P = I (no row swaps needed)
• The PA = LU framework as a flowchart for any invertible matrix
• Why numerical libraries (MATLAB, NumPy, LAPACK) use PA = LU with partial pivoting for stability

━━━━━━━━━━━━━━━━━━━━━━━━
SOURCE MATERIALS
The source materials for this video are from    • 5. Transposes, Permutations, Spaces R^n