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The matrix I described above is 2 by 2, but has just a single eigenvector. So you're wrong about n x n matrices having n eigenvalues.
That's why algebraic multiplicities (2, in my example) are needed, rather than geometric multiplicities (1, in my example). This is true of any non-normal matrix, except in the trivial case with eigenvalue 1, where raising to different powers makes no difference.
No, the problem of missing an eigvec, and hence by your semantics missing an eigval, is not true of any non-normal matrix, only special non-generic ones (an infinitesimal perturbation will give full basis of eigvecs). e.g. [[2, epsilon][1,2]] has two eigvecs for ep \neq 0.
Excuse me - I meant to say degenerate non-normal matrix. Again, with the caveat that the degeneracy is not for eigenvalue 1.
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