Results 1 to 2 of 2

Math Help - Jordan Decomposition to Schur Decomposition

  1. #1
    Junior Member
    Joined
    Mar 2009
    Posts
    46

    Jordan Decomposition to Schur Decomposition

    Let A be a complex or real square matrix. Suppose we have a Jordan decomposition A = XJX^{-1}, where X is non-singular and J is upper bidiagonal. Show how you can obtain a Schur Decomposition from a Jordan Decomposition.


    Schur Decomposition: A = QTQ*, where Q is unitary/orthogonal and T is upper triangular with the eigenvalues of A on the diagonal.


    I'm really not sure at all what to do. Because Q is orthogonal, Q*= Q^{-1}. I'm not sure if that plays in somehow.

    I've been trying to use SVDs or QR decompositions of X or J to get there, but I've had no luck. Does anyone have any suggestions? Thank you so much.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Jul 2008
    Posts
    81
    QR should be right?

    X=QR yields XJX^{-1}=QRJR^{-1}Q^{-1}=Q(RJR^{-1})Q^* but R,J,R^{-1} are all upper triangular, so their product is as well, giving the required decomposition. The eigenvalue thing is a consequence of properties of triangular matrices and similarity preserving eigenvalues.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. QR decomposition, completing Q
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: January 16th 2012, 01:26 AM
  2. decomposition of r.v.
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: November 5th 2011, 03:48 PM
  3. LU-decomposition
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 14th 2011, 02:28 AM
  4. QR Decomposition
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: May 11th 2010, 08:36 PM
  5. Jordan form and primary decomposition
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: April 17th 2010, 12:34 PM

Search Tags


/mathhelpforum @mathhelpforum