PLU-Decomposition of a Matrix Explained | Linear Algebra

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If a matrix requires a row swap to get it into row echelon form, then it does not have an LU-decomposition. For such a matrix A, we may permute its rows before factoring, and then an LU decomposition will exist. In interchanging the rows, we essentially multiply by an elementary matrix Q (or a product of elementary matrices) to carry out the row swaps. This product QA will not require row swaps and so has an LU decomposition. By letting P=Q^-1, we solve for A and get A=PLU, a PLU-factorization of the matrix A. Every square matrix has such a decomposition. As with LU decompositions, PLU decompositions can be used to solve systems of linear equations, which we will show. #linearalgebra

LU Decomposition Explained:    • LU-Decomposition of a Matrix Explained | L...  

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0:00 Intro
0:48 PLU-Decompositions
2:11 How to Solve a Linear System
2:52 Solving a Linear System with PLU
4:11 Find LU Decomposition
7:19 The PLU Factorization
7:57 Solving
11:00 Conclusion