Exercise 10.B.4

Proof. This is follows easily from the definitions. If $\lambda$ is an eigenvalue of $T$, or $T_{}$

C$,then$c$\lambda$ is an eigenvalue of $cT$, or ${(cT)}_{}$

C$,withequalmultiplicity.Let$${\lambda }_1,\dots ,{\lambda }_n$ denote the eigenvalues of $T_{}$

C$,repeatedaccordingtoitsmultiplicity.Then$c${\lambda }_1,\dots ,c{\lambda }_n$ are the eigenvalues of ${(cT)}_{}$

C$andwehave$

□