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Math Help - Eigenvectors

  1. #1
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    Eigenvectors

    I know how to get the eigenvalues, but hiow do i get the eigenvectors?

    \left(\begin{array}{cc}3&1\\1&3\end{array}\right)

    (3- \lambda)(3- \lambda) - 1

    lambda = 2 and 4

    how do i do the eigenvectors of this?
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  2. #2
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    Quote Originally Posted by adam_leeds View Post
    I know how to get the eigenvalues, but hiow do i get the eigenvectors?

    \left(\begin{array}{cc}3&1\\1&3\end{array}\right)

    (3- \lambda)(3- \lambda) - 1

    lambda = 2 and 4

    how do i do the eigenvectors of this?
    For \lambda=2, the matrix \begin{bmatrix}3-\lambda&1\\1&3-\lambda\end{bmatrix} becomes \begin{bmatrix}1&1\\1&1\end{bmatrix}. You then have to solve the system of equations \begin{bmatrix}1&1\\1&1\end{bmatrix}\begin{bmatrix  }x\\y\end{bmatrix} = \begin{bmatrix}0\\0\end{bmatrix}, which reduces to x+y=0. This has the solution x=1, y=–1. So \begin{bmatrix}1\\-1\end{bmatrix} is an eigenvector for the eigenvalue 2.

    Now do the same thing for \lambda=4.
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  3. #3
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    Or, just apply the definition of "eigenvector".

    If 2 is an eigenvalue, then there exist a non-zero eigenvector, \begin{pmatrix}x \\ y\end{pmatrix} such that

    \begin{pmatrix}3 & 1 \\ 1 & 3\end{pmatrix}\begin{pmatrix}x \\ y\end{pmatrix}= \begin{pmatrix}3x+ y\\ x+ 3y\end{pmatrix}= 2\begin{pmatrix}x \\ y\end{pmatrix}

    That gives the two equations 3x+ y= 2x and x+ 3y= 2y. Both of those reduce to the equation y= -x. That is
    \begin{pmatrix}x \\ y\end{pmatrix}= \begin{pmatrix}x \\ -x\end{pmatrix}= x\begin{pmatrix}1 \\ -1\end{pmatrix}.

    An eigenvector corresponding to eigenvalue 2 is \begin{pmatrix}1 \\ -1\end{pmatrix}.

    With eigenvalue 4 instead of 2, those equations become 3x+ y= 4x and x+ 3y= 4y, both of which reduce to y= x. An eigenvector corresponding to eigenvalue 4 is \begin{pmatrix}1 \\ 1\end{pmatrix}.
    Last edited by Opalg; May 5th 2010 at 01:31 PM. Reason: sorted out LaTeX
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