# When is possible to use an orthogonal matrix to put in Jordan form a matrix?

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?

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