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On number of negative eigenvalues of a matrix

  1. Aug 11, 2010 #1
    1. The problem statement, all variables and given/known data
    When trying to solve a question about parameter independence of certain aspects of the Jacobian of a real valued function on a manifold I came to the point where I have to show the following:

    Let A be a matrix, J be the Jacobian of an orthogonal transformation (I suppose we can assume non-reflection) then define B to be B= JTAJ, where A and B are real symmetric. I have to show that both B and A have same number of negative eigenvalues.


    3. The attempt at a solution
    This problem I think translates into following:

    Both A and B are real symmetric so suppose their diagonal forms are respectively D1 and D2 (ie matrices which have eigenvalues for the diagonal entries). Then there is a unitary transformation U such that D2 = UTD1U (an be shown by direct calculation). I have to now show that the number of negative diagonal entries on each D is the same. It seems logical at first because U is an orthogonal matrix but direct calculation does not yield the answer. Moreover when I try to put it into a geometric setting (i.e rotation of vectors) it seems wrong. Am I doing something wrong? edit: I think this might be a wrong approach though because I haven't used the fact that there is a Jacobian as a part of U

    Thanks.
     
    Last edited: Aug 11, 2010
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  3. Aug 11, 2010 #2

    lanedance

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    if you could show [tex] J^T = J^{-1} [/tex] it would follow pretty quickly
     
  4. Aug 12, 2010 #3
    I dont think it is the case, it is just the Jacobian of any orthogonal coordinate transformation (not the transformation itsself)
     
  5. Aug 12, 2010 #4

    lanedance

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    for a 2D rotation
    [tex]u = x cos(\theta)+ y sin(\theta)[/tex]
    [tex]v = x sin(\theta)- y cos(\theta)[/tex]

    the jacobian is
    [tex]J(x,y) = \begin{pmatrix}
    \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y }\\
    \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y}
    \end{pmatrix} [/tex]

    which is looking very similar to the transformation itself...maybe within a T... if you agree, could you generalise that?
     
    Last edited: Aug 12, 2010
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