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Thread: Eigenvectors

  1. #1
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    Unhappy Eigenvectors

    The matrix i have is |3 4| |4 -3| How do I find the corresponding eigenvectors? 2x2 matrix. 1st row 3 4, 2nd is 4 -3.
    I have done the first step of the process, and found eigenvalues of +-5. Now how do I find the eigenvectors? I have aksed this question before, and I have recieved help which I already know how to do. When I substitue in the eigenvalues to find the eigenvectors, I get x1=0 and x2=0, so there must be some other way to find the eigenvector. If possible, find a final solution to the problem so I know the method works. Thanks
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  2. #2
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    Quote Originally Posted by LooNiE View Post
    The matrix i have is |3 4| |4 -3| How do I find the corresponding eigenvectors? 2x2 matrix. 1st row 3 4, 2nd is 4 -3.
    I have done the first step of the process, and found eigenvalues of +-5. Now how do I find the eigenvectors? I have aksed this question before, and I have recieved help which I already know how to do. When I substitue in the eigenvalues to find the eigenvectors, I get x1=0 and x2=0, so there must be some other way to find the eigenvector. If possible, find a final solution to the problem so I know the method works. Thanks
    $\displaystyle \lambda = 5$: $\displaystyle \left( \begin{array}{cc}
    3 & 4 \\

    4 & -3\end{array}\right)$ $\displaystyle \left( \begin{array}{c}
    x \\
    y \end{array}\right)$ $\displaystyle = 5 \left( \begin{array}{c}
    x \\
    y \end{array}\right)$

    Therefore:

    $\displaystyle 3x + 4y = 5x \Rightarrow x = 2y$ .... (1)

    $\displaystyle 4x - 3y = 5y \Rightarrow x = 2y$ .... (2)

    Therefore the corresponding eigenvector has the form $\displaystyle \left( \begin{array}{c}
    2y \\
    y \end{array}\right)$ $\displaystyle = y \left(\begin{array}{c}
    2 \\
    1 \end{array}\right)$ and so the eigenvector is $\displaystyle \left( \begin{array}{c}
    2 \\
    1 \end{array}\right)$.

    Do the same thing for $\displaystyle \lambda = -5$.
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