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A^Tx=b given LU factorization for A

  1. Dec 10, 2011 #1
    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
  2. jcsd
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