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Thread: Eigenvalues and Eigenvectors help

  1. #1
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    Eigenvalues and Eigenvectors help

    Suppose A is an N N matrix with eigenvectors vi, i = 1, 2, 3 N and correspondingeigenvalues λi, i = 1, 2, 3 N.

    (a) Prove that 5/10 = λ1λ2λ3 λN



    (b) Consider the characteristic polynomial det(A − λI) for the matrix A.
    Show that the coefficient aN−1 of the λN−1term is given byaN−1 = (−1)N−1(λ1 + λ2 + λ3 + + λN )
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  2. #2
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    Re: Eigenvalues and Eigenvectors help

    Quote Originally Posted by GossipGoat View Post
    Suppose A is an N N matrix with eigenvectors vi, i = 1, 2, 3 N
    and correspondingeigenvalues λi, i = 1, 2, 3 N.

    (a) Prove that 5/10 = λ1λ2λ3 λN [/quote]
    Do you mean "prove that the product of all the eigenvalues is 1/2"? If so why write 1/2 as "5/10". In any case, that is not true. You can construct an N by N matrix with any n eigenvalues you want- so their product can be any number.



    (b) Consider the characteristic polynomial det(A − λI) for the matrix A.
    Show that the coefficient aN−1 of the λN−1term is given byaN−1 = (−1)N−1(λ1 + λ2 + λ3 + + λN )
    Do you mean the λ^(N-1) (λ to the N-1 power)? Also is "(-1)N-1 supposed to be (-1)^(N-1)?
    Note that (x- a)(x- b)= x^2-(a+ b)x+ ab, (x- a)(x- b)(x- c)= x^3- (a+ b+ c)x^2+ (ab+ ac+ bc)x- abc, etc.
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  3. #3
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    Re: Eigenvalues and Eigenvectors help

    Hi

    I just reread my original post and realised that the formatting totally screwed it up

    Here is the question

    Eigenvalues and Eigenvectors help-untitled.png

    Sorry about that
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