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Math Help - another question about eigenvalues and linear operators

  1. #1
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    another question about eigenvalues and linear operators

    hi, i have another question, last one - so i hope you won't mind:

    i need to prove that:

    S and T are two linear operators that can switch sides (ST = TS) over a vector space V.
    lambda is a eigenvalue of S. the group W is defined as Ker ((lambda)I - S).
    prove that W is contained in T(W).

    thanks in advance to anyone who can help.
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  2. #2
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    Quote Originally Posted by vonflex1 View Post
    hi, i have another question, last one - so i hope you won't mind:

    i need to prove that:

    S and T are two linear operators that can switch sides (ST = TS) over a vector space V.
    lambda is a eigenvalue of S. the group W is defined as Ker ((lambda)I - S).
    prove that W is contained in T(W).

    thanks in advance to anyone who can help.
    are you sure it's not the other way round, i.e. T(W) \subseteq W? that's a trivial result: if w \in W, then S(w)=\lambda w and so ST(w)=TS(w)=\lambda T(w), which means T(w) \in W. so T(W) \subseteq W.
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