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Math Help - I need help for this matrix differential equation!

  1. #1
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    I need help for this matrix differential equation!

    Let A be a real 3 3 matrix, u and v linearly independent vectors in R3 such that Au = u
    and Av = v. Suppose w is a vector in R3 such that Aw = w + u + v.

    (i) Find all eigenvalues of A.
    (ii) Solve the differential system x′ = Ax.



    For (ii) I know that 1 is definitely an eigenvalue, is there any other eigenvalues apart from 1? Thanks in advance!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: I need help for this matrix differential equation!

    Quote Originally Posted by alphabeta89 View Post
    Let A be a real 3 3 matrix, u and v linearly independent vectors in R3 such that Au = u
    and Av = v. Suppose w is a vector in R3 such that Aw = w + u + v.

    (i) Find all eigenvalues of A.
    (ii) Solve the differential system x′ = Ax.
    Prove that B=\{u,u+u,w\} is a basis of \mathbb{R}^3. Then, \begin{Bmatrix}Au=u\\A(u+v)=u+v\\Aw=(u+v)+w\end{ma  trix} which implies that J=\begin{bmatrix}{1}&{0}&{0}\\{0}&{1}&{1}\\{0}&{0}  &{1}\end{bmatrix} is the canonical Jordan form of A.
    Thanks from alphabeta89
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    Re: I need help for this matrix differential equation!

    Very nice.
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: I need help for this matrix differential equation!

    Quote Originally Posted by HallsofIvy View Post
    Very nice.
    Nice to hear that.
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