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Math Help - eigenvalue/eigenvector question

  1. #1
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    eigenvalue/eigenvector question

    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...
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by ruprotein View Post
    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
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