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Consider the symmetric matrices associated to orthogonal reflections of the Euclidean plane. Since a line comes back to itself after a half-turn, you get a family indexed by $[0,\pi]$ describing a closed path in the set of symmetric matrices. A corresponding eigenvector of eigenvalue $1$ and norm $1$, deformed continuously goes into its opposite. This path can thus not be lifted into a continuous part formed by eigenvectors.