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Obviously, the magnitude of the orthogonal matrix is 1, which is easy to prove.. However, I wonder how can one prove that the eigenvalue of an orthogonal matrix is $-1$, if the determinant of this matrix is $-1$?

Thanks a lot.

Thanks for the help that I can prove it now from the conjugate pairs point of view.

Is there a method by proving that $\det(I+A)=0$?

Davide Giraudo
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Jack2019
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1 Answers1

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Hint: nonreal eigenvalues of real orthogonal matrices must occur in conjugate pairs, and the product of a conjugate pair of complex number is ...

user1551
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  • And the product of a conjugate pair of complex number is the magnitude, which is positive number, therefore, there has to be at least one negative eigenvalue, and therefore has to be -1. Thnaks a lot ! – Jack2019 Apr 22 '13 at 20:24
  • If determinant is -1? I still don't get it – user21795 Nov 27 '16 at 11:53
  • @user21795 There is a theorem which says the product of the eigenvalues equals the determinant. – mdcq Apr 11 '18 at 21:02