I heard that all real eigenvalues of an orthogonal matrix are either $1$ or $-1$. Why is that?

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    $\begingroup$ The real eigenvalues are! There may be complex eigenvalues! $\endgroup$ – Will Jagy May 11 '16 at 21:08
  • $\begingroup$ Related: math.stackexchange.com/questions/653133/… $\endgroup$ – user174622 May 11 '16 at 21:09
  • $\begingroup$ Changed the question. I am asking about real eigenvalues. $\endgroup$ – jackskis May 11 '16 at 21:09
  • $\begingroup$ Thank you @YagnaPatel, but the question you link to doesn't answer my question $\endgroup$ – jackskis May 11 '16 at 21:11
  • $\begingroup$ For future reference, putting the problem statement only in the title is poor practice. The body of the Question should be used to give a full statement of the problem and some context: Why is the problem interesting to you? How do you relate the outcome to the assumptions of the problem? Are there special cases you were able to solve? Any of these elements of context will help Readers to understand your difficulty and respond in ways more likely to help you. $\endgroup$ – hardmath May 12 '16 at 0:40

Hint: If $x$ is an eigenvector and $M$ is an orthogonal matrix, consider $\|Mx\|$.

  • $\begingroup$ I am not sure where you are going. Any additional hints would be appreciated. $\endgroup$ – jackskis May 11 '16 at 21:16
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    $\begingroup$ Maybe think about it for more than $3$ minutes. Try something, flip through the textbook, then tell me you're stuck. $\endgroup$ – Ben Grossmann May 11 '16 at 21:19
  • $\begingroup$ Well, I know that $\det(M) = \pm 1$. Does knowing that help my cause? $\endgroup$ – jackskis May 11 '16 at 21:44
  • $\begingroup$ Not so much. Note that $\|Mx\|=\|x\|$, but $\|Mx\|=\|\lambda x\|$. $\endgroup$ – Ben Grossmann May 12 '16 at 0:02
  • $\begingroup$ The key property of orthogonal real matrices is that multiplication by one preserves the length of a vector. Check this out in your textbook. $\endgroup$ – hardmath May 12 '16 at 0:50

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