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Math Help - General Questions about Eigenvalues and eigenvectors

  1. #1
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    General Questions about Eigenvalues and eigenvectors

    Hi. I have some questions regarding eigen-related things.

    1) If A & B are similar matrices do they have the same eigenvectors? (I know they have the same eigenvalues...) If not, can someone please give an example?

    2) A is a square matrix. Do A and A^t have the same eigenvalues? Do A and A^t have the same eigenvectors? If not can someone please give an example?

    Thanks!
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    For 1 & 2;

    if A & B are similar matrices they have same characteristic polynomial so...

    use my above argument to prove q. 2.
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  3. #3
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    For some reason I can't relate eigenvectors to characteristic polynomials.
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  4. #4
    Senior Member roninpro's Avatar
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    You might not be able to. It is possible to have two matrices with the same eigenvalues but different eigenvectors.

    Try to find an example for yourself.
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  5. #5
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    The sticky about proofs will show examples.
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  6. #6
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    I'm afraid AlsoSprachZarathustra misunderstood when you said "(I know they have the same eigenvalues...)".

    If A and B are similar matrices then B= P^{-1}AP for some invertible P. If x is an eigenvector of A, with eigenvalue \lambda, let y= P^{-1}x. Then By= P^{-1}APy= P^{-1}APP^{-1}x = P^{-1}Ax= P^{-1}\lambda x= \lambda P^{1}x= \lambda y.

    Thus, if similar matrices have the same eigenvalues but NOT, in general, the same eigenvectors.
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  7. #7
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    Thus, if similar matrices have the same eigenvalues but NOT, in general, the same eigenvectors.
    Does this hold true also for it's A^t
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  8. #8
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    Quote Originally Posted by jayshizwiz View Post
    Does this hold true also for it's A^t
    If you would go to the sticky in the forum, you would have your answers since I have pdf showing these scenarios and many more available for download.
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  9. #9
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    O, I just noticed where the stickies are lol... Thanks.
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