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For all orthogonally diagonalizable real matrices, or symmetric real matrices, are all eigenvalues distinct? What would be the proof it is so, or if not, what would be the proof?

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Before asking such a question you might try to think of the most obvious example of diagonal real matrices. In general, while eigenvalues often happen to be distinct, there are very few natural conditions that force them to be distinct. – Marc van Leeuwen Dec 15 '12 at 9:47

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Nope. For instance, consider the identity matrix. All the eigenvalues are $1$.

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