Linear Algebra Done Right · Problem 13

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · nilpotent-jordan-null-space

Problem

Suppose

T \in \mathcal{L}(V)

is nilpotent. Prove that if

v_1, \ldots, v_n

are vectors in

V

and

m_1, \ldots, m_n

are nonnegative integers such that

T^{m_1}v_1, \ldots, Tv_1, v_1, \ldots, T^{m_n}v_n, \ldots, Tv_n, v_n \text{ is a basis of } V

and

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

then

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

is a basis of null

T

. This exercise shows that

n = \dim \text{null}\, T

. Thus the positive integer

n

that appears above depends only on

T

and not on the specific Jordan basis chosen for

T