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Jacobi method convergence for hpd matrices

  1. Jun 1, 2012 #1
    1. The problem statement, all variables and given/known data
    Let A be a squared, hermitian positive definite matrix. Let D denote the diagonal matrix composed of the diagonal elements of A, i.e. D = diag((A)11,(A)22,...(A)nn).

    Prove that if the Jacobi iterative method converges for A, then 2D - A must also be hermitian positive definite.


    2. Relevant equations

    The Jacobi iterative method has the iteration matrix J = inv(D) * (E + E*), where A = D - E - E* is the standard decomposition of the matrix to a diagonal matrix, upper triangular matrix and lower triangular matrix.

    Also, an iterative method converges if and only if the iteration matrix has a spectral radius lower than 1, i.e. r(J) < 1.

    3. The attempt at a solution

    I have tried several approaches. I was able to prove the inverse claim, i.e. that if 2D - A is hpd, then the Jacobi method must converge. Also, I observed that the iteration matrix for the Jacobi method of 2D - A is just -J, so the method converges for both A and 2D - A.
     
  2. jcsd
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