Let T be a linear operator on a vector space V over the field F, and let g(t) be a p;olynomail with coefficients from F. Prove that if x is an eigecvector of T with corresponding eiogenvalue λ then g(T)x= g(λ)x.
Prove...
I'm note sure this needs proof.
Let g(t)=sum_{i=0,..n}} f_i t^i, then
g(T)x = sum_{i=0,..n}} f_i T^i x
but T^i x= lambda^i x as x is an eigen vector corresponding to eigen value lambda, so:
g(T)x = sum_{i=0,..n}} f_i lambda^i x =g(lambda) x
(that T^i x= lambda^i x can be proven by induction if it is not blindingly
obvious to you )
RonL