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QR Factorization

  1. Mar 18, 2015 #1
    1. The problem statement, all variables and given/known data
    Consider an invertible n x n matrix A. Can you write A as A=LQ, where L is a lower triangular matrix and Q is orthogonal? Hint: Consider the QR factorization of #A^T#.


    2. Relevant equations
    For QR factorization, Q is orthogonal and R is upper triangular.

    3. The attempt at a solution
    If we consider the hint, then we can write:
    ##A^T=S*U## where S is orthogonal matrix and U is some upper triangular matrix.
    ##(A^T)^T=U^T*S^T##; transpose of upper triangular matrix U is some lower triangular matrix L
    ##A=L*S^T##

    Here is where I get lost. I don't know how to show that S^T=Q. Could someone please give me a hint?
     
    Last edited: Mar 18, 2015
  2. jcsd
  3. Mar 18, 2015 #2

    Mark44

    Staff: Mentor

    Your # symbols are cluttering up your work, making it harder to read than it should be. If you trying to use LaTeX, use two # at the beginning and two more at the end.

    Regarding your question, S is orthogonal, right. What about its transpose, ST? Isn't that orthogonal as well?
     
  4. Mar 18, 2015 #3
    Ah yes! I think I overthought this problem.
     
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