How does LU decomposition work?

Question

I'm interested in the algorithm of LU decomposition in order to solve a LSE like $Ax=b$, where $A$ is a square matrix.

My question is: When I compute $PA=LU$ do I also need to interchange rows in $L$ whenever I interchange rows in $A$? It is clear, that I get $P$ by interchanging rows in $Id$ whenever I interchange rows in $A$, but is $L$ also affected?

Is it right that after that I need to solve $Lz = Pb$ and $Ux = z$?

Answer 1

To solve $Ax = b$, if $P A = LU$, then $LUx = Pb$:

a) compute $P A = LU$ factorization, saving $P$ info

b) solve $Ly = P b$, using forward substitution

c) solve $Ux = y$, using backward substitution.