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Let $A$ be a real square matrix. Say some of its eigenvalues are unitary complex numbers, but it also has other eigenvalues that are not unitary (they may be real or complex). Must the unitary eigenvalues always be non-defective?

My reasoning is the following: let $S$ be the subspace that is spanned by all eigenvectors corresponding to unitary eigenvalues. Then the projection of $A$ to $S$ is a unitary matrix and therefore all its eigenvalues are non-defective. However, I'm confused as to whether it is possible for the original $A$ to have generalized eigenvectors outside of the space $S$.

Any pointers appreciated!

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    $\begingroup$ What if $A$ is the companion matrix of $(x^2+1)^2$? $\endgroup$ – Angina Seng Jul 29 '20 at 15:53

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