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Homework Help: Matrix Proof Help

  1. Sep 24, 2010 #1
    Hello,
    I need a hint on how to begin this proof, please.

    Prove that if A is a projection matrix, A2=A, then I + A is invertible and
    (I + A) -1 = I - [tex]\frac{1}{2}[/tex]A.
     
  2. jcsd
  3. Sep 24, 2010 #2

    hunt_mat

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    I would write B=a*I+b*A and compute B*(I+A) and (I+A)B and see for what values of a and b I find an inverse. Note that A^2=A, so there are no higher powers in the series expansion of the inverse.
     
  4. Sep 24, 2010 #3

    Dick

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    If I+A and I-A/2 are supposed to be inverses then their product should be I, right? Is it?
     
  5. Sep 24, 2010 #4
    Well, [tex]I-A/2[/tex] is a matrix. Why don't you try to see what you get if you multiply it by [tex]I+A[/tex]? Maybe it will tell you something?
     
  6. Sep 24, 2010 #5
    I tried multiplying (I + A) and (I - A/2) and indeed it equals I. That's it!? That proves it ?
     
  7. Sep 24, 2010 #6

    Dick

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    Sure that proves it. If MN=I then M=N^(-1) and N=M^(-1). It's the definition of 'inverse'.
     
  8. Sep 24, 2010 #7
    thank you
     
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