From the left-hand eigenvectors, λ λ β β ′ | Φ Φ β β ′ [ J C ] ⟩ ⟩ = ⟨ ⟨ Φ Φ β β ′ [ D K ] | ψ ψ ′ ⟩ ⟩ = ∑ ∑ i , j , α α , γ γ ( Γ Γ β β γ γ ′ [ D ] j ) ∗ ∗ Θ Θ α α γ γ i j ( λ λ γ γ ) 2 λ λ α α | α α i ⟩ ⟩ after expressing them in the basis { | i α α ⟩ ⟩ } , the Γ Γ [ C ] 's are: | Φ Φ ′ [ J C ] ⟩ ⟩ = ∑ ∑ i , α α Γ Γ α α β β ′ [ C ] i λ λ α α | α α i ⟩ ⟩ .