Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 15D / a / Solution

The operator $D$ vanishes on the orthogonal complement of $S=\mathrm{span}\{|{\alpha }_0⟩,|{\alpha }_1⟩\}$, so it has rank at most two. The states are distinct rays, hence they are linearly independent and $S$ has dimension two. In an orthonormal basis of $S$, its determinant isThus its two eigenvalues on $S$ are nonzero and have opposite signs. All remaining eigenvalues are zero.