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Math Help - Eigenvalue Method

  1. #1
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    Eigenvalue Method

    Apply the eigenvalue method to find a general solution to:

    x1' = x1 + 2x2 + 2x3
    x2' = 2x1 + 7x2 + x3
    x3' = 2x1 + x2 + 7x3

    Here's what Ive done:

    The matrix form of the system is:

    x' = 1 2 2
    .......2 7 1
    .......2 1 7

    Then the book says to do this:

    A - &I = 1-&....2......2
    ..............2....7-&....1
    ..............2.....1.....7-&

    I can't follow the rest. Can someone show how to do the rest? Thanks
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  2. #2
    Master Of Puppets
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    Next thing to do is solve  \displaystyle |A-\lambda I| = 0


     \displaystyle (1-\lambda)\times [(7-\lambda)(7-\lambda)- 1\times 1] - 2[2(7-\lambda)-1(7-\lambda)] +2[2\times 1-(7-\lambda)\times 2 ]=0
    Then follow the rest of this example

    Pauls Online Notes : Differential Equations - Real Eigenvalues
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  3. #3
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    Ok I get 3 eigenvalues: 0,6,9

    Case 1:
    I plug 0 into the matrix from the original post and notcie that its a singular matrix b/c the det is 0. In a 2x2 matrix, this means I can arbitrary choose a and solve for b. How does this work for my 3x3 matrix? Thanks
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  4. #4
    A Plied Mathematician
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    Well, what do you get for your row reduction? Incidentally, you can type up matrices as follows:

    \begin{bmatrix}<br />
1-\lambda &2 &2\\<br />
2 &7-\lambda &1\\<br />
2 &1 &7-\lambda<br />
\end{bmatrix}, and augmented matrices this way:

    \left[\begin{matrix}1 &0 &0\\ 0 &1 &0\\ 0 &0 &1\end{matrix}\right \left|\begin{matrix}4\\5\\6\end{matrix}\right]

    Just double-click on the matrices to see how I entered them, and then enclose that code in math tags.
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