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Homework Help: Invertible matrix

  1. Apr 22, 2010 #1
    Suppose A is a Hermitian positive definite matrix split into [tex] A = C + C^{*} + D [/tex] where [tex] D[/tex] is also Hermitian positive definite.
    We show that [tex]B=C+ \omega ^{-1} D[/tex] is invertible. Consider the iteration [tex] x_{n+1} = x_{n} + B^{-1} (b-Ax_{n})[/tex] , with any initial iterate [tex]x_{0}[/tex] . Prove that [tex]x_{n}[/tex] converges to [tex]x= A^{-1}b [/tex] whenever [tex]0< \omega < 2[/tex].

    I suppose to show invertibility, we need to show [tex] det(B) \neq 0[/tex]. But I am not sure how to continue.
  2. jcsd
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