F=ℝ: Normal matrix with real eigenvalues but not diagonalizable

Bipolarity
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I am going through Friedberg and came up with a rather difficult problem I can't seem to resolve.
If ## F = ℝ ## and A is a normal matrix with real eigenvalues, then does it follow that A is diagonalizable? If not, can I find a counterexample?

I'm trying to find a counterexample, by constructing a matrix A that is normal and has real eigenvalues, but is not diagonalizable. It is giving me some problems! Any ideas?

If I did not have the assumption that A has real eigenvalues, the rotation matrix would suffice as a counterexample.

If I had the complex field instead of the real field, then easily A is diagonalizable since normality implies orthogonal diagonalizibility in a complex inner product space.

BiP
 
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Just a hint: there is no counterexample, look for proof.
 

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