# A^Tx=b given LU factorization for A

## Homework Statement

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).

## Homework 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

## 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.

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