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Homework Help: Eigenvalue for Orthogonal Matrix

  1. Nov 28, 2011 #1
    1. The problem statement, all variables and given/known data

    Let Q be an orthogonal matrix with an eigenvalue [itex]λ_{1}[/itex] = 1 and let x be an eigenvector belonging to [itex]λ_{1}[/itex]. Show that x is also an eigenvector of [itex]Q^{T}[/itex].

    2. Relevant equations

    Qx = λx where x [itex]\neq[/itex] 0

    3. The attempt at a solution

    [itex]Qx_{1} = x_{1}[/itex] for some vector [itex]x_{1}[/itex]

    [itex](Qx_{1})^{T} = x_{1}^{T}Q^{T}[/itex]

    I'm kind of stuck with how to start this problem, as I'm not sure what I've done is even starting down the right path. Can anyone give me a nudge in the right direction?
  2. jcsd
  3. Nov 28, 2011 #2


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    What's (Q^T)(Q) if Q is real orthogonal?
  4. Nov 28, 2011 #3
    Thank you. I didn't realize what an orthogonal matrix was (yikes!). Once I did the proof fell right out of the definition.
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