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Homework Help: Eigenvalue of a matrix.

  1. Jan 13, 2012 #1
    Proof involving nonsingular matrices.

    1. The problem statement, all variables and given/known data

    If (I + A) is nonsingular, prove that (I - A)(I + A)-1 = (I + A)-1(I - A), and hence (I - A)/(I + A) is defined for the matrix.

    I've proved it like this:

    Let (I - A)(I + A)-1 = A, and (I + A)-1(I - A) = B.
    B-1 = (I - A)-1(I + A)
    B-1A = I
    Premultiplying by B, we get A = B.

    Is this proof correct?
    Last edited: Jan 13, 2012
  2. jcsd
  3. Jan 13, 2012 #2
    Re: Proof involving nonsingular matrices.

    You don't know that I-A is invertible. So (I-A)-1 might not exist.
  4. Jan 14, 2012 #3
    Re: Proof involving nonsingular matrices.

    Oh, I forgot to mention that A is known to be skew-symmetric. So, (I - A)T = (I + A), which is nonsingular. And since a matrix is nonsingular iff its transpose is nonsingular, we could assume that (I - A)-1 exists.

    I can't seem to think beyond this point. If there's still an error somewhere in the proof, could you please point to it?
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