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Math Help - Eigenvectors/values Question

  1. #1
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    Eigenvectors/values Question

    (a) [8 marks] Let V be a real vector space and let T: V\rightarrow V be a linear transformation.
    Define the terms eigenvalue and eigenvector of T. Show that if v_1, . . . , v_n are eigenvectors
    for distinct eigenvalues \lambda_1,...,\lambda_n then the set of v_1,...,v_n is linearly independent.

    (b) [12 marks] Now let v_1,...,v_n be a basis of eigenvectors with distinct eigenvalues \lambda_1,...,\lambda_n. Prove that for any linear map S:V\rightarrow V with ST = TS, v_i is also an eigenvector of S for each i = 1, . . . , n.
    Deduce that an nn matrix that commutes with every diagonal nn matrix must
    be diagonal itself.

    So far I have managed part a), but am stuck on the first part of b). I have tried taking TS(v) and then using the eigenvalues and the fact that S and T are linear to find something, but I can't get anything of the form S(v)=av (a constant). Could someone start me off?
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Eigenvectors/values Question

    Quote Originally Posted by worc3247 View Post
    (a) [8 marks] Let V be a real vector space and let T: V\rightarrow V be a linear transformation.
    Define the terms eigenvalue and eigenvector of T. Show that if v_1, . . . , v_n are eigenvectors
    for distinct eigenvalues \lambda_1,...,\lambda_n then the set of v_1,...,v_n is linearly independent.

    (b) [12 marks] Now let v_1,...,v_n be a basis of eigenvectors with distinct eigenvalues \lambda_1,...,\lambda_n. Prove that for any linear map S:V\rightarrow V with ST = TS, v_i is also an eigenvector of S for each i = 1, . . . , n.
    Deduce that an nn matrix that commutes with every diagonal nn matrix must
    be diagonal itself.

    So far I have managed part a), but am stuck on the first part of b). I have tried taking TS(v) and then using the eigenvalues and the fact that S and T are linear to find something, but I can't get anything of the form S(v)=av (a constant). Could someone start me off?

    Hint for (a): use induction.

    For (b): use this this theorem:

    Matrix M_n is similar to diagonal Matrix D iff there are exist n linear independent eigenvectors, diagonal elements of D are eigenvalues and D=P^{-1}MP , where P is matrix with eigenvectors on the columns.
    Last edited by Also sprach Zarathustra; June 15th 2011 at 05:46 AM.
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    A Plied Mathematician
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    Re: Eigenvectors/values Question

    Quote Originally Posted by worc3247 View Post
    (a) [8 marks] Let V be a real vector space and let T: V\rightarrow V be a linear transformation.
    Define the terms eigenvalue and eigenvector of T. Show that if v_1, . . . , v_n are eigenvectors
    for distinct eigenvalues \lambda_1,...,\lambda_n then the set of v_1,...,v_n is linearly independent.

    (b) [12 marks] Now let v_1,...,v_n be a basis of eigenvectors with distinct eigenvalues \lambda_1,...,\lambda_n. Prove that for any linear map S:V\rightarrow V with ST = TS, v_i is also an eigenvector of S for each i = 1, . . . , n.
    Deduce that an nn matrix that commutes with every diagonal nn matrix must
    be diagonal itself.

    So far I have managed part a), but am stuck on the first part of b). I have tried taking TS(v) and then using the eigenvalues and the fact that S and T are linear to find something, but I can't get anything of the form S(v)=av (a constant). Could someone start me off?
    If these questions are for marks, it is against forum policy knowingly to help with such questions.

    Thread closed. PM me if you wish to discuss it further.
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    A Plied Mathematician
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    Re: Eigenvectors/values Question

    Thread re-opened based on PM from OP'er.
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