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Urgent: eigenvector problem

  1. Jun 15, 2005 #1
    Hi

    I have this here matrix

    [tex]A = \left[ \begin{array}{ccc} 2 & 1 & 0 \\ 0 & 1 & 0 \\ 3 & 3 & 0 \end{array} \right][/tex]

    I calculate the eigenvalues and get (2,1,-1)

    Next I calculate the eigenvectors and get (1,0,1) and (-1,1,0) and (0,0,0)

    My professor says my third eigenvector is wrong and it should (0,0,1)

    My calculation:


    [tex]A = \left[ \begin{array}{ccc} (2-(-1) & 1 & 0 \\ 0 & (1-(-1) & 0 \\ 3 & 3 & 1-(-1) \end{array} \right]
    = \left[ \begin{array}{ccc} 3 & 1 & 0 \\ 0 & 2 & 0 \\ 3 & 3 & 0 \end{array} \right][/tex]

    Then according to the theorem regarding eigenvectors:

    [tex]\left[ \begin{array}{ccc} 3 & 1 & 0 \\ 0 & 2 & 0 \\ 3 & 3 & 0 \end{array} \right] \left[ \begin{array}{c} v_1 \\ v_2 \\ v_3 \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right] [/tex]

    then

    [tex]3v_1 + v_2 = 0[/tex]

    [tex] 2v_2 = 0[/tex]

    [tex]3v_1 + 3 v_2 = 0[/tex]

    Is my calculations correct ??

    sincerley and best regards,

    Fred
     
  2. jcsd
  3. Jun 15, 2005 #2
    First of all, the zero vector is never an eigenvector (even though it might behave like one). If x is an eigenvector "belonging" to the eigenvalue -1, then Ax = -x, or equivalently (A + I)x = 0. The right-most entry in the bottom row of this equation:

    [tex]\left[ \begin{array}{ccc} 3 & 1 & 0 \\ 0 & 2 & 0 \\ 3 & 3 & 0 \end{array} \right] \left[ \begin{array}{c} v_1 \\ v_2 \\ v_3 \end{array} \right] = \left[ \begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right] [/tex]

    is wrong, there should be a 1 there instead of a 0.
     
    Last edited: Jun 15, 2005
  4. Jun 15, 2005 #3
    Check your eigenvalues again. I get [tex]\lambda = 0,1,2[/tex]
     
  5. Jun 15, 2005 #4
    Hi and many thanks for Your answer.

    I have a second question I hope You can answer for me.

    Finding a matrix P which is invertible and which complies with P^-1 AP ? ?

    Isn't P then span of the three eigenvector ???

    Sincerley and Best Regards,

    Fred

     
  6. Jun 15, 2005 #5

    Hi again

    thats because I typed my matrix wrong

    [tex]A = \left[ \begin{array}{ccc} 2 & 1 & 0 \\ 0 & 1 & 0 \\ 3 & 3 & -1 \end{array} \right][/tex]

    Sincerley

    Fred
     
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