TITLE: Why a complex symmetric matrix is not diagonalizible?
QUESTION [6 upvotes]: I know an Hermitian matrix is diagonalizable, and similarly a real symmetric matrix is diagonalizable, but what's wrong in a complex symmetric matrix. 
Why does the Gram-Schmidt process fail?

REPLY [6 votes]: As Chris Godsil and Dietrich Burde pointed out it's because $\langle x,y \rangle =x^*y=0$ which is the orthogonality condition on complex vectors does not imply that $x^Ty=0$ which is the complex symmetry condition. 
So the Gram Schmidt process actually will produce orthogonal vectors, but they will not be able to diagonalize the matrix.