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Question about adjoint transformations- is this a valid proof

  1. Jan 3, 2012 #1
    1. The problem statement, all variables and given/known data
    Q is an invertible self-adjoint linear transformation on an inner product space V. Suppose Q is positive definite. I have already shown that inv(Q) is self-adjoint, that all eigenvalues of Q are positive, so there exists S s.t. S^2 = Q.
    Now suppose P is any self-adjoint linear transformation on V. I have shown inv(S)*P*inv(S) is also self adjoint. The bit I'm trying to do is: deduce or prove otherwise that there are linearly independant vectors e1...en and scalars a1...an such that for i between 1 and n:
    Pei = aiQei

    2. Relevant equations
    Spectral theorem.

    3. The attempt at a solution
    I have been staring at this for ages- I've done this but I'm not sure at all that its valid so if its complete rubbish please let me know!

    inv(S)*P*inv(S) is self-adjoint so by the spectral theorem there exists an orthonormal (and therefore linearly independent) basis of eigenvectors e1...en with associated eigenvalues a1...an s.t

    inv(S)*P*inv(S)ei = aiei for i = 1..n

    (possibly dodgy step coming up!)
    Now consider the transformation inv(S)*P*inv(S) restricted to span(ei)
    On this subspace of V, inv(S)*P*inv(S) = ai*I
    pre-multiply by S, post multiply by S
    to get P = ai*Q on this span
    so Pei = aiQei

    Is that correct?
    Thanks.
     
  2. jcsd
  3. Jan 4, 2012 #2
    Can anyone help? So sorry for bumping but even a 'yes' or 'no' answer as to whether the proof is valid would be extremely useful.
    Thanks again.
     
  4. Jan 4, 2012 #3

    Dick

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    Well, no. It can't be a valid proof. You can only restrict the action of S to span(ei) if ei is also an eigenvector of S. That much should be pretty clear. I'm still not clear whether your final conclusion is true or not though. I'll tell you one thing I think is true though. Pick Q=[[1,0],[0,4]] and P=[[0,1],[0,1]]. I think you can find two independent vectors such that Pei = aiQei. But they aren't orthogonal. Clearly I'm still fishing around with this problem.
     
    Last edited: Jan 4, 2012
  5. Jan 5, 2012 #4
    Thanks.. dodgy step was indeed dodgy then. Yeah the vectors being linearly independent should obviously be easier to prove than being orthogonal (which may not be true at all) so I will look at that. Just struggling at the moment to relate inv(S)Pinv(S) and Q since there is no guarentee (infact its unlikely surely?) that Q is diagonal over the same basis as inv(S)Pinv(S) is. I have in my notes that two matrices are simultaneously diagonalisable iff they commute but here that seems to be what I'm trying to prove anyway.
     
  6. Jan 5, 2012 #5

    Dick

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    Ok, think I see it finally. As you said, you have inv(S)*P*inv(S)ei = aiei. S is invertible. So pick vi=inv(S)ei. Since the ei are a basis, so are the vi (but not necessarily orthogonal). The vi are the vectors you want. Do you see it?
     
  7. Jan 5, 2012 #6
    yep I do thank you! Was getting rather frustrated because the rest of the question I could prove if I had proved this bit first. Thanks again.
     
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