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Homework Help: Orthogonal Matrix

  1. Dec 10, 2009 #1
    1. The problem statement, all variables and given/known data

    Let l be an eigenvalue of an orthogonal matrix A, where l = r + is. Prove that l * conj(l) = r^2 + s^2 = 1.

    2. Relevant equations



    3. The attempt at a solution

    I am really confused on where to go with this one.

    I have Ax = A I x = A A^T A x = l^3 x

    and Ax = l x so l x = l^3 x

    l = l^3

    l^2 = 1
    l = 1 or -1

    But I can't really figure out what to do from here, am I even on the right track?

    Thanks for the help
     
  2. jcsd
  3. Dec 10, 2009 #2
    Hi, i am just studying linear algebra (the final is coming next week, so stressed!!).

    I think the question about l*conjugate(l) = r^2+s^2 = 1 just mean the length of l is 1

    But i think if Matrix with complex entries, say M, is orthogonal means M is unitary. So the length of every eigenvalue is 1

    since unitary matrix won't change vector's langth so its eigenvalues' length is always 1
     
  4. Dec 10, 2009 #3

    D H

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    Staff Emeritus
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    phrygian cannot use this fact; he/she has to prove it.

    phrygian, the cube route isn't really going to help here. You went one step too far.

    You know that [tex]\mathbf A \vec x = \lambda \vec x[/tex]. The conjugate transpose of the right-hand side is [tex](\lambda \vec x)^* = \vec x^*\lambda^*[/tex]. What is the matrix product of this conjugate transpose with [tex] \lambda \vec x[/tex]?
     
  5. Dec 11, 2009 #4
    Is there a way to do this without using conjugate transposes? The book has a hint that says first show that ||Ax|| = ||x|| for any vector x, but i just cant seem to get past that first step
     
  6. Dec 12, 2009 #5

    HallsofIvy

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    What is the precise definition of "orthogonal" matrix? Some texts use that a matrix, Q, is orthogonal if and only if <Qu, v>= <u, Qv> for all vector u and v (< u, v> is the inner product). From that it is not too hard to show that ||Qv||= ||v||.
     
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