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


closed as off-topic by Leucippus, Jean-Claude Arbaut, Giovanni, Shailesh, Chris Godsil May 12 '16 at 0:46

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Jean-Claude Arbaut, Giovanni, Shailesh, Chris Godsil
If this question can be reworded to fit the rules in the help center, please edit the question.

  • 1
    $\begingroup$ 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/… $\endgroup$ – Yagna Patel 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
  • 1
    $\begingroup$ Maybe think about it for more than $3$ minutes. Try something, flip through the textbook, then tell me you're stuck. $\endgroup$ – Omnomnomnom 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 $\|Mx\|=\|\lambda x\|$. $\endgroup$ – Omnomnomnom 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

Not the answer you're looking for? Browse other questions tagged or ask your own question.