Linear Mathematics | Part IB, 2001

Let VV be a vector space over R\mathbb{R}. Let α:VV\alpha: V \rightarrow V be a nilpotent endomorphism of VV, i.e. αm=0\alpha^{m}=0 for some positive integer mm. Prove that α\alpha can be represented by a strictly upper-triangular matrix (with zeros along the diagonal). [You may wish to consider the subspaces ker(αj)\operatorname{ker}\left(\alpha^{j}\right) for j=1,,mj=1, \ldots, m.]

Show that if α\alpha is nilpotent, then αn=0\alpha^{n}=0 where nn is the dimension of VV. Give an example of a 4×44 \times 4 matrix MM such that M4=0M^{4}=0 but M30M^{3} \neq 0.

Let AA be a nilpotent matrix and II the identity matrix. Prove that I+AI+A has all eigenvalues equal to 1 . Is the same true of (I+A)(I+B)(I+A)(I+B) if AA and BB are nilpotent? Justify your answer.

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