Linear Algebra | Part IB, 2004

(i) Let VV be an nn-dimensional vector space over C\mathbb{C} and let α:VV\alpha: V \rightarrow V be an endomorphism. Suppose that the characteristic polynomial of α\alpha is Πi=1k(xλi)ni\Pi_{i=1}^{k}\left(x-\lambda_{i}\right)^{n_{i}}, where the λi\lambda_{i} are distinct and ni>0n_{i}>0 for every ii.

Describe all possibilities for the minimal polynomial and prove that there are no further ones.

(ii) Give an example of a matrix for which both the characteristic and the minimal polynomial are (x1)3(x3)(x-1)^{3}(x-3).

(iii) Give an example of two matrices A,BA, B with the same rank and the same minimal and characteristic polynomials such that there is no invertible matrix PP with PAP1=BP A P^{-1}=B.

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