This exercise is related to the phase-transition limits of (16.33). (a) Let (x) = x{1+ (x1) 1}.

This exercise is related to the phase-transition limits of (16.33).

(a) Let ψ(x) = x{1+γ (x−1)

−1}. Show that ψ(x) is strictly increasing for x > 1 +

√

γ .

(b) Show that for the top eigenvalues of Σ that exceed the threshold, 1 +

√

γ , the limits of their corresponding ˆλj (in terms of the ordering in eigenvalues) follow the same order, that is, if λj > λk > 1 +

√

γ, we have lim ˆλj > lim ˆλk.