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Math Help - What is the relationship of the eigenvectors between 2 similar matrices?

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    What is the relationship of the eigenvectors between 2 similar matrices?

    Let A and B be similar matrices. Prove that there is a nice relationship between the eigenvectors of A and the eigenvectors of B
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  2. #2
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    Quote Originally Posted by nicolem1051 View Post
    Let A and B be similar matrices. Prove that there is a nice relationship between the eigenvectors of A and the eigenvectors of B
    Vectors or Values?

    Similar matrices have the same eigenvalues but not necessarily the same eigenvectors.
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    vectors - yea that's what I can't figure out!
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    I suppose you could say that the relationship between the eigenvectors is that they share the same eigenvalue. If you're looking for how to prove that the eigenvalues are the same, I have written a short proof here:

    det(A-xI)=det(inv(P)BP-xI)
    =det(inv(P)BP-inv(P)xIP)
    =det(inv(P)(B-xI)P)
    =det(B-xI)

    Thus, as the characteristic polynomials are the same, the eigenvalues are the same.

    I've put a formatted version up: Maths homework help
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  5. #5
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    If A and B are similar matrices, then there exist an invertible matrix P such that A= P^{-1}BP.

    Let v be an eigenvector of A corresponding to eigenvalue \lambda. Then Av= (P^{-1}BP)v= \lambda v. Multiplying on both sides by P, (BP)v= P\lambda v so B(PV)= \lambda (PV).

    That is, if v is an eigenvalue of A corresponding to eigenvalue \lambda then Pv is an eigenvalue of B also corresponding to eigenvalue \lambda.

    Of course, we can think of P as a "change of basis" matrix. That is, A and B represent the same linear transformation, written in different bases. The eigenvectors of A and B are representations of the same vector, written in different bases.
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