Linear Algebra Done Right · Problem 23

Operators on Real Vector Spaces — Sheldon Axler

Mathematics — Linear Algebra · complexification-eigenvalues

Problem

Suppose

\mathbf{F} = \mathbf{R}

T \in \mathcal{L}(V)

, and

\lambda \in \mathbf{C}

. (a) Show that

u + iv \in G(\lambda, T_{\mathbf{C}})

if and only if

u - iv \in G(\bar{\lambda}, T_{\mathbf{C}})

. (b) Show that the multiplicity of

\lambda

as an eigenvalue of

T_{\mathbf{C}}

equals the multiplicity of

\bar{\lambda}

as an eigenvalue of

T_{\mathbf{C}}

. (c) Use (b) and the result about the sum of the multiplicities (8.25) to show that if

\dim V

is an odd number, then

T_{\mathbf{C}}

has a real eigenvalue. (d) Use (c) and the result about real eigenvalues of

T_{\mathbf{C}}

(Exercise 17 in Section 5A) to show that if

\dim V

is an odd number, then

T

has an eigenvalue (thus giving an alternative proof of 5.34). See Exercise 33 in Section 3B for the definition of the complexification

T_{\mathbf{C}}