Why are the real eigenvalues of an orthogonal matrix always 1 or -1?

Question

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

Related: http://math.stackexchange.com/questions/653133/eigenvalues-in-orthogonal-matrices — , May 11 '16 at 21:09
Thank you @YagnaPatel, but the question you link to doesn't answer my question — jackzellweger, May 11 '16 at 21:11
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. — hardmath, May 12 '16 at 00:40
Ok thank you @hardmath! I will keep that in mind in the future. I appreciate it. — jackzellweger, May 12 '16 at 00:44

score 2 · Answer 1 · answered May 11 '16 at 21:13

2

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

answered May 11 '16 at 21:13

Ben Grossmann

I am not sure where you are going. Any additional hints would be appreciated. – jackzellweger May 11 '16 at 21:16
1

Maybe think about it for more than $3$ minutes. Try something, flip through the textbook, then tell me you're stuck. – Ben Grossmann May 11 '16 at 21:19
Well, I know that $\det(M) = \pm 1$. Does knowing that help my cause? – jackzellweger May 11 '16 at 21:44
Not so much. Note that $|Mx|=|x|$, but $|Mx|=|\lambda x|$. – Ben Grossmann May 12 '16 at 00:02
The key property of orthogonal real matrices is that multiplication by one preserves the length of a vector. Check this out in your textbook. – hardmath May 12 '16 at 00:50

1 Answers1