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Math Help - DE system

  1. #1
    Junior Member
    Joined
    Apr 2010
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    DE system

    Please help me with this one:


    y'=\begin{pmatrix}<br />
2 &  2&0 \\ <br />
 2& 2 &0 \\ <br />
 0& 0 & 4<br />
\end{pmatrix} y

    Thanks.
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  2. #2
    MHF Contributor

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    Find the eigenvalue of the matrix. (Hint: one of the eigenvalues is 4. What are the eigenvalues of \begin{pmatrix}2 & 2 \\ 2 & 2 \end{pmatrix}?)

    Find the corresponding eigenvectors. Fortunately, for this problem there are three independent eigenvectors so the matrix is "diagonalizable" (every symmetric matrix is diagonalizable). That is, if P is the matrix having the eigenvectors as columns, P^{-1}AP= D where A is the given matrix and P is the diagonal matrix having the eigenvalues on the main diagonal.

    Multiply the entire equation by P to get Py'= (Py)'= PAy. Let X=Py so that y= P^{-1}X. Then the equation becomes X'= PAP^{-1}Py= DX. X'= DX is easy to solve and then y= P^{-1}X.
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