Show that the following matrix has no real eignvalues, and thus no eigenvectors. Interpret your result geometrically.
Are you sure you have written the matrix correctly?
Charikar is right. The matrix has real eigenvalues and eigenvectors. The matrix is a reflection - the eigenvectors are the bisectors of the axes.
Your question would make sense if the matrix had been
as that would have been a rotation, which does not have real eigenvectors.