Linear Algebra Done Right · Problem 5E.10

Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · commuting-operators

Problem

Give an example of two commuting operators

S, T

on a finite-dimensional real vector space such that

S + T

has an eigenvalue that does not equal an eigenvalue of

S

plus an eigenvalue of

T

and

ST

has an eigenvalue that does not equal an eigenvalue of

S

times an eigenvalue of

T

. This exercise shows that 5.81 does not hold on real vector spaces.