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I know that if I have a symmetrical matrix defined on $R$, it is always diagonalisable and I can always find beetwen the matrix of its eigenspaces an orthogonal matrix. While if I have a non diagonalisable matrix, of course I can put it in Jordan normal form, however how the matrix must be to be able to find out an orthogonal matrix of its generalized eigenvectors?

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  • $\begingroup$ I'm not aware of any nice criteria in this direction unless the Jordan normal form is diagonal. $\endgroup$ – Qiaochu Yuan Dec 4 '18 at 20:27

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