Results 1 to 3 of 3

Math Help - Hermitian positive definite matrix and invertibility

  1. #1
    Member
    Joined
    Feb 2009
    Posts
    98

    Hermitian positive definite matrix and invertibility

    Suppose A is a Hermitian positive definite matrix split into  A = C + C^{*} + D where  D is also Hermitian positive definite.
    We show that B=C+ \omega ^{-1} D is invertible. Consider the iteration  x_{n+1} = x_{n} + B^{-1} (b-Ax_{n}) , with any initial iterate x_{0} . Prove that x_{n} converges to x= A^{-1}b whenever 0< \omega < 2.

    I suppose to show invertibility, we need to show  det(B) \neq 0. But I am not sure how to show that. Also for the convergence, do we show lim_{n \rightarrow \infty } \left\| x_{n}-x \right\|= 0 ? If yes, how?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    You want to show that B is invertible, but B^{-1} appears in the expression which you plan on using to do that... Are you sure you copied the problem properly?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Feb 2009
    Posts
    98
    yes, the problem is copied properly. I think we need to first show that B is invertible (i.e. B inverse exists) to be able to use the expression for  x_{n+1} .
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof of positive definite matrix
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: March 11th 2010, 01:41 AM
  2. Hermitian positive definite matrice
    Posted in the Algebra Forum
    Replies: 0
    Last Post: October 1st 2009, 12:00 PM
  3. 2x2 Positive Definite matrix.
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: September 10th 2009, 05:39 PM
  4. positive definite matrix
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 26th 2008, 10:57 AM
  5. Replies: 0
    Last Post: June 4th 2008, 03:39 PM

Search Tags


/mathhelpforum @mathhelpforum