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Well we don't disagree then except on semantics. I think of n x n matrix always having n eigenvalues and Det is their product. I think that's very standard. But you're using a different convention.
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.
It's semantics. As I said, different conventions. You consider it an eigenvalue only if associated w/ eigenvector, so if matrix is missing an eigenvec, you consider it missing an eigenvalue. I think of the eigenvalues as the solutions to the characteristic Eq., n x n always has n
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