 Solution

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

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

For

D = P^{- 1} D_{0} P

, conjugating

U = D^{- 1} T

gives

P U P^{- 1} = D_{0}^{- 1} (D_{0} + N) = I + D_{0}^{- 1} N .

Because

D_{0}^{- 1}

commutes with

N

, and

N^{n} = 0

(D_{0}^{- 1} N)^{n} = D_{0}^{- n} N^{n} = 0.

Thus

U - I

is similar to a nilpotent matrix, and is itself nilpotent. Therefore $U$ is a unipotent matrix.

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