I know this is kind of stupid, but does anyone know that is there any theorem actually proved the uniqueness of eigenvector?
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As J. M. noted, any eigenvector may be multiplied by a scalar, hence it's not unique strictly speaking. Plus in case of degeneracy, situation where several eigenvectors correspond to the same eigenvalue, any linear combination of them is the eigenvector as well.
So that eigenvectors for a specific eigenvalue actually span a corresponding sub-space. This sub-space however is strictly defined.