 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/ib/paper-2/15g/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

U

N = U - I

Uv = λ v

for a nonzero eigenvector, then

N^{m} v = (λ - 1)^{m} v = 0

for some

m

λ = 1

Conversely, suppose the only eigenvalue is

1

. Over

C

(t - 1)^{n}

(U - I)^{n} = 0

, so

U - I

is nilpotent. Thus $U$ is unipotent if and only if its only eigenvalue is one.

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