(i) Show that MM† and M†M are Hermitian matrices. (ii) Show that their eigenvalues are non-negative. Hint: this does not require a complex calculation. Simply show that they satisfy the defining property of positive semi-definite matrices. (iii) Show that both MM† and M†M obey the requirement that density matrices have unit trace. Hint: use the fact that the state of the system plus bath is correctly normalized, ⟨Ψ∣Ψ⟩=1.