 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2014/11/ii/solution

Put

p = ϵ + δ + η

and

q = ϵδ + δη + ηϵ

. Direct expansion gives the characteristic polynomial

det (λ I - M) = (λ + ϵ) (λ + η) (λ + δ) - ϵηδ = λ (λ^{2} + p λ + q) .

The quadratic formula therefore gives

K = - \frac{ϵ + δ + η}{2}, W = ϵδ + δη + ηϵ, λ_{\pm} = K \pm K^{2} - W .

Since

p, q > 0

, both nonzero eigenvalues have negative real part. They need not be real; equal rates give a complex-conjugate pair.

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