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Math Help - eigenvalues

  1. #1
    Newbie
    Joined
    Sep 2008
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    Wink eigenvalues

    Hi,
    Plese help!
    Suppose T is a linear map and dim(Im(T))=k. Prove that T has at most k+1 distinct eigenvalues.
    Thank you!
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  2. #2
    Junior Member
    Joined
    May 2008
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    for every distinct eigenvalue you have at least one eigenvector. let the set A contain one eigenvector for every non zero eigenvalue. eigenvectors for distinct eigenvalues are independent. because they are eigenvectors for nonzero eigenvalues you have
    span(A) = span(T(A)) - so  dim(Im(T)) \ge dim(span(T(A))) = dim(span(A)) = |A|
    so there are at most dim(Im(T)) non zero eigenvalues, or dim(Im(T))+1 eigenvalues
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