eigenvalues of matrices in AA

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Thanks for reporting, this is now trac ticket 18071 .There seems that eigenvectors_right() is not implemented for the real algebraic field AA. The workaround is to define your matrix in QQbar:

sage: M = M.change_ring(QQbar)
sage: M.eigenvectors_right()
[(1*I, [
  (1, -1*I)
  ], 1), (-1*I, [
  (1, 1*I)
  ], 1)]

It also works in ZZ:

sage: M = M.change_ring(ZZ)
sage: M.eigenvectors_right()
[(-1*I, [(1, 1*I)], 1), (1*I, [(1, -1*I)], 1)]