In the answer here: https://math.stackexchange.com/a/833622/155881, a word of caution is raised around the possibility that a matrix could have the Eigen vectors corresponding to distinct Eigen values orthogonal to each other and yet non-diagonalizable because one of the Eigen values has multiple Eigen vectors, not all independent. It is proved that this isn't possible with symmetric matrices. But to drive home the point, is it possible with non-symmetric real matrices? If so, is there an example of such a matrix?
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2$\pmatrix{1&0&0\ 0&0&1\ 0&0&0}$. – user1551 Apr 16 '23 at 07:45
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1You meant multiple eigenvalues, not eigenvectors. – Ryszard Szwarc Apr 16 '23 at 08:24
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1Any matrix in Jordan canonical form has this property. – eyeballfrog Apr 16 '23 at 11:27