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# Rotation matrix always has eigenvalue 1 Watch

1. I've been trying to disprove the statement:

"If A is a 2x2 real matrix without real eigenvalues then A is a rotation matrix."

However I have read in numerous places that rotation matrices always have 1 as an eigenvalue, so the above statement would not hold, because if A does not have any real eigenvalues, then it can't have 1 as an eigenvalue and hence can't be a rotation matrix, however I am struggling to prove this in the 2x2 case.

My (poor) attempt:

The 2x2 rotation matrix is always of the form:

So it has characteristic equation

where x is our eigenvalue.

which reduces down to

which when solving with the quadratic formula we get

however I can't see how x = 1 is always a solution to this? In fact, I'm fairly certain it's not, any advice at all would be great.
2. (Original post by TheIrrational)
...
I refer you to wiki

1 is an eigenvalue of a 3x3 rotation matrix, but not a 2x2 rotation matrix.
3. (Original post by ghostwalker)
I refer you to wiki

1 is an eigenvalue of a 3x3 rotation matrix, but not a 2x2 rotation matrix.
Ah god, I overlooked matrix size when I was reading through these. Thanks, that would explain why it doesn't work for me.

Edit: Was easy enough to disprove by counter example anyway.

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Updated: April 8, 2013
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