# eigenvalues of (real?) orthogonal matrices have magnitude 1

• May 9th 2011, 07:05 PM
HD09
eigenvalues of (real?) orthogonal matrices have magnitude 1
Hi all, just wanted to make sure I have this right -

My lecture notes say that the eigenvalues of orthogonal matrices have magnitude 1. When I looked through the proof it seemed to me like the statement should have said *real* orthogonal matrices.

So I played around a little bit and I think I came up with a counterexample to show the eigenvalues of any orthogonal matrix don't have magnitude 1:

(LaTeX isn't co-operating, sorry)

Q = [2i, sqrt{5} ; sqrt{5}, -2i]

is orthogonal but has eigenvalues 2i & -1/(2i), which don't have magnitude 1...

All good? Anything I missed?

Thanks a lot
• May 9th 2011, 07:19 PM
Tinyboss
Orthogonal matrices by definition have real entries. The complex analogue is called "unitary".
• May 9th 2011, 08:25 PM
HD09
Thanks!
• May 10th 2011, 02:01 AM
Ackbeet
In addition, I'd like to point out that this question has probably been asked on MHF more often than any other question. Do please use Google and even our search routines to try to get an answer that way before asking your question here.