Exercise 9.A.11

Proof. From the definitions, it is easy to see that

C^2 + bT_C + cI) = 0.

Therefore $z^2+bz+c$ is polynomial multiple of the minimal polynomial of $T_{}$

C$(see8.46).Notethat,becausethispolynomialhasrealcoefficients,iteitherhastworealrootsortwononrealroots.Thus$T$hasaneigenvalueifandonlyif$T_C$hasarealroot(see9.10),whichhappensifandonlyif$z^2 + bz + c$hasarealroot(bythepreviousreasoningand8.49),whichhappensifandonlyif$b^2 $\ge 4c$. □