Let be a matrix with a real, simple eigenvalue. Prove that there exists a neighborhood of in which all matrices have at least one real eigenvalue.

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# Keeping it real

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Let be a matrix with a real, simple eigenvalue. Prove that there exists a neighborhood of in which all matrices have at least one real eigenvalue.

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A small perturbation of the coefficients of a real polynomial with a simple root results in a polynomial with a simple root.

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