Kunze Linear Algebra · Problem 7.1.5

Rational and Jordan Forms

Mathematics — Methods · nilpotent-cyclic-vectors

Problem

Let VV be an nn-dimensional vector space over the field FF, and let NN be a nilpotent linear operator on VV. Suppose Nn−1≠0N^{n-1} \neq 0, and let α\alpha be any vector in VV such that Nn−1α≠0N^{n-1}\alpha \neq 0. Prove that α\alpha is a cyclic vector for NN. What exactly is the matrix of NN in the ordered basis {α,Nα,…,Nn−1α}\{\alpha, N\alpha, \ldots, N^{n-1}\alpha\}?

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