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Math Help - Rangesand Eigenvalues

  1. #1
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    Rangesand Eigenvalues

    Hey, I have a question here that states:

    a) Find T belongs to L(Cē) [C being the set of complex numbers] with two distinct eigenvalues such that dim rangeT = 1.
    b) Suppose T belongs to L(V) and dim rangeT = K. Prove that T has at most k+1 distinct eigenvalues.

    Could anyone give me an idea on how to approach this question. Any help would be greatly appreciated. Thanks.
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  2. #2
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    Quote Originally Posted by GreenDay14 View Post
    Hey, I have a question here that states:

    a) Find T belongs to L(Cē) [C being the set of complex numbers] with two distinct eigenvalues such that dim rangeT = 1.
    b) Suppose T belongs to L(V) and dim rangeT = K. Prove that T has at most k+1 distinct eigenvalues.

    Could anyone give me an idea on how to approach this question. Any help would be greatly appreciated. Thanks.
    If \lambda_1 and \lambda_2 are distinct eigenvalues, then the respective eigenvectors are independent. So suppose eigenvalues \lambda_1 and \lambda_2 have eigenvectors u and v, respectively. Then u and v, separately, span different subspaces and so Tu= \lambda_1u and Tv= \lambda_2v span different subspaces unless one of the eigenvalues is 0!
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