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).
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
The Attempt at a Solution
Haha...I figured it out...
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.