16.-Linear-Algebra-by-Kenneth-Hoffman · Problem 7.1.5

The Rational and Jordan Forms

Mathematics — Linear Algebra · nilpotent-operators-cyclic-vectors

Problem

Let

V

be an

n

-dimensional vector space over the field

F

, and let

N

be a nilpotent linear operator on

V

. Suppose

N^{n-1} \neq 0

, and let

\alpha

be any vector in

V

such that

N^{n-1}\alpha \neq 0

. Prove that

\alpha

is a cyclic vector for

N

. What exactly is the matrix of

N

in the ordered basis

\{\alpha, N\alpha, \ldots, N^{n-1}\alpha\}