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Math Help - similar matrices and eigenvalues

  1. #1
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    similar matrices and eigenvalues

    Prove using the definition of eigenvalues that similar matrices have the same eigenvalues.

    I have already shown this using the characteristic polynomial but I have no idea how to do it this way.


    Any help would be greatly appreciated!
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  2. #2
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    Quote Originally Posted by pseudonym View Post
    Prove using the definition of eigenvalues that similar matrices have the same eigenvalues.

    I have already shown this using the characteristic polynomial but I have no idea how to do it this way.


    Any help would be greatly appreciated!

    Supose A\sim B\Longrightarrow A=P^{-1}AP\Longleftrightarrow PAP^{-1}=B for some invertible matrix P.
    Now let Av=\lambda v\,\,and\,\,Pv=w\Longrightarrow P^{-1}w=v, so then:

    Bw=PAP^{-1}w=PAv=P(\lambda v)=\lambda Pv=\lambda w\Longrightarrow \lambda is an eigenvalue of B

    Tonio
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