Question

Problem 4. Give an example of a linear operator T on a finite-dimensional vector space such that T is not nilpotent, but zero is the only eigenvalue of T. Characterize all such operators.

Problem 5. Let A be an n × n matrix whose characteristic polynomial splits, γ be a

cycle of generalized eigenvectors corresponding to an eigenvalue λ, and W be the subspace spanned

by γ. Define γ′ to be the ordered set obtained from γ by reversing the order of the vectors in γ.

(a)Prove that [TW]γ′ =([TW]γ)^t

(b) Let J be the Jordan canonical form of A. Use (a) to prove that J and J^t are similar.

(c) Use (b) to prove that A and A^t are similar.

Problem 4. (5 points) Give an example of a linear operator T on a finite-dimensional vector space such that T is not nilpotent, but zero is the only eigenvalue of T. Characterize all such operators. Problem 5. (5 points) Let A be an n x n matrix whose characteristic polynomial splits, 7 be a cycle of generalized eigenvectors corresponding to an eigenvalue X, and W be the subspace spanned by 7. Define y' to be the ordered set obtained from by reversing the order of the vectors in %. (a) Prove that (Twly = (Twl)

Answer 1

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