 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1/5/i/solution

Let

v_{1}, \dots, v_{n}

V = (v_{1} \dots v_{n})

A V = V D

, where

D = diag (λ_{1}, \dots, λ_{n})

. Hence

V^{- 1} A V = D,

B = P^{- 1} A P

, then

P^{- 1} v_{j}

are linearly independent eigenvectors of

B

with the same eigenvalues. Explicitly, with

W = P^{- 1} V

W^{- 1} B W = V^{- 1} P (P^{- 1} A P) P^{- 1} V = D .

Thus matrix similarity preserves diagonalizability.

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