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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.2012-12-15

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