 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2016/paper-1/5/c/ii/solution

For a possibly complex eigenvector

v \neq = 0

with

A v = λ v

, invertibility of

I - A

gives

(I - A)^{- 1} v = (1 - λ)^{- 1} v

. Therefore

N v = \frac{1 + λ}{1 - λ} v, ν = \frac{1 + λ}{1 - λ} .

Thus every eigenvector of

A

remains an eigenvector of

N

λ = i t

with

t

real, so

ν = e^{2 i a r c t a n t}

lies on the unit circle, consistently with orthogonality.

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