I am going through Friedberg and came up with a rather difficult problem I can't seem to resolve.(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

**Physics Forums | Science Articles, Homework Help, Discussion**