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We study square matrices  F ( α ) over ℂ with  α ∈ℝ, where the eigenvalues depend on the parameter  α  but not the eigenvectors, and vice versa, where the eigenvectors depend on the parameter  α  but not the eigenvalues. We derive sufficient conditions for such properties. Applications to Lie groups and spin systems are provided. Both normal and nonnormal matrices are investigated.

Eigenvalue Problem, Spin Systems, Lie Groups, and Parameter Dependence