Show that the following matrix has no real eignvalues, and thus no eigenvectors. Interpret your result geometrically.
[0 -1]
[-1 0]
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
[0 -1]
[1 0]
as that would have been a rotation, which does not have real eigenvectors.