Linear Algebra Done Right · Problem 8A.11

Operators on Complex Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · generalized-eigenvectors

Problem

Suppose that

T \in \mathcal{L}(V)

. Prove that there is a basis of

V

consisting of generalized eigenvectors of

T

if and only if the minimal polynomial of

T

equals

(z - \lambda_1) \cdots (z - \lambda_m)

for some

\lambda_1, \ldots, \lambda_m \in \mathbf{F}

. Assume

\mathbf{F} = \mathbf{R}

because the case

\mathbf{F} = \mathbf{C}

follows from 5.27(b) and 8.9. This exercise states that the condition for there to be a basis of

V

consisting of generalized eigenvectors of

T

is the same as the condition for there to be a basis with respect to which

T

has an upper-triangular matrix (see 5.44). Caution: If

T

has an upper-triangular matrix with respect to a basis

v_1, \ldots, v_n

V

, then

v_1

is an eigenvector of

T

but it is not necessarily true that

v_2, \ldots, v_n

are generalized eigenvectors of

T