 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ii/paper-4/40e/a/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

M x = (A - B) x

and suppose

H x = λ x

, so

- B x = λ M x

. Put

a = x^{*} A x > 0

and

m = x^{*} M x

. Since

A = M - (- B)

a = (1 - λ) m

. Also

x^{*} (A - B - B^{T}) x = 2 Re m - a = a \frac{1 - ∣ λ ∣ ^{2}}{∣1 - λ ∣ ^{2}} .

Positive definiteness makes this positive, hence

∣ λ ∣ < 1

for every eigenvalue and

ρ (H) < 1

 New to topics? Read the docs here!