# A^Tx=b given LU factorization for A

1. Dec 10, 2011

### dbkats

1. The problem statement, all variables and given/known data

Suppose you are given the LU factorization for some nxn square matrix A. Assume A is non-singular. This factorization is a result of partial pivoting. Can you use this factorization to solve A^Tx=b for x (given A and b).

2. Relevant equations

A^T is the transpose of matrix A.
PA = LU is the assumed factorization of A with partial pivoting
Since P is a permutation matrix, P^T=P^-1

3. The attempt at a solution

Haha...I figured it out...

PA=LU
A = (P^T)LU
A^T = (U^T)(L^T)P

A^Tx = (U^T)(L^T)Px = b

Then let Px = y

A^Tx = (U^T)(L^T)y = b

U^T is then lower-triangular, L^T is unit-upper-triangular. Therefore I can solve for y in the usual way, and then figure out what x is based on the permutation matrix P.

Last edited: Dec 10, 2011