Linear Algebra Done Right · Problem 9

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · nilpotent-jordan-basis-existence

Problem

Suppose

T \in \mathcal{L}(V)

is nilpotent. Explain why there exist

v_1, \ldots, v_n \in V

and nonnegative integers

m_1, \ldots, m_n

such that (a) and (b) below both hold. (a)

T^{m_1}v_1, \ldots, Tv_1, v_1, \ldots, T^{m_n}v_n, \ldots, Tv_n, v_n

is a basis of

V

. (b)

T^{m_1+1}v_1 = \cdots = T^{m_n+1}v_n = 0