# LU decomposition permutation matrix

Hi can you help me with the following;

Let $A$ be an $n\times n$ matrix and have $LU$ decomposition with lower and upper triangular matrices. Let $P =\{e_n,e_{n-1},\ldots,e_1\}$ where $e_i$ is a unit vector i.e. $P$ is the permutation matrix. Prove that $PAP$ has $UL$ factorization with $U$ upper triangular and $L$ lower triangular matrix i.e. $PAP = UL$

I thought easily $PAP = (PL)(UP)$ are upper and lower parts but they are NOT!

Please help.

Thank you!!

• Please note: Write P=\{e_n,e_{n-1},\ldots,e_1\} and put that whole thing in math mode, and it looks like this: $P=\{e_n,e_{n-1},\ldots,e_1\}$. There's no need for the complicated way you did it. – Michael Hardy Jan 23 '13 at 3:44
• @Michael yeah, you know you are not talking Yobo there, right? – adam W Jan 23 '13 at 4:04
• I didn't know that, but from the edit history I see that it was Maisam Hedyelloo who introduced the pointlessly complicated TeX code. – Michael Hardy Jan 23 '13 at 4:07

## 1 Answer

Hint: $PAP = PLUP = (PLP)(PUP)$.