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Math Help - Finding eigenvectors

  1. #1
    Senior Member I-Think's Avatar
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    Finding eigenvectors

    The matrix A, given by
    A=\left(\begin{array}{ccc}7&-4&6\\2&2&2\\-3&4&-2\end{array}\right)

    has eigenvalues 1,2,4. Find the set of corresponding vectors.

    Every time I perform these operations I only get my answers as 0's. I'll geve an example.

    x(A-\delta{I})=0 taking \delta as 1

    \left(\begin{array}{ccc}6&-4&6\\2&1&2\\-3&4&-3\end{array}\right) \left(\begin{array}{c}x\\y\\z\end{array}\right)=\l  eft(\begin{array}{c}0\\0\\0\end{array}\right)

    Solving
    6x-4y+6z=0
    2x+y+2z=0
    -3x+4y-3z=0

     -x+5y-z=0
     x=5y-z

     30y-6z-4y+6z=0
     y=0
    And so on. Could someone please direct me as to what I am doing wrong?
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  2. #2
    Super Member Gamma's Avatar
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    Iowa City, IA
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    Just by inspection, you can see that an eigenvector that will work is <1,0,-1>.
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  3. #3
    Super Member
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    You're solving the equations wrong (note that the coefficients of x and z will always vanish together if we try to sum up one equation to another) , the solutions for that eigenvalue should be x=-z
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