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Real and Zero eigenvalues

  1. May 18, 2016 #1
    1. The problem statement, all variables and given/known data
    [tex]
    \frac{d\vec{Y}}{dt}
    =
    \begin{bmatrix}
    2 & 4 \\
    3 & 6
    \end{bmatrix}
    \vec{Y}[/tex]
    Find the eigenvalues and eigenvectors
    2. Relevant equations


    3. The attempt at a solution
    I found the eigenvectors to be
    [tex]
    \vec{v_1} =
    \begin{bmatrix}
    2 \\
    1
    \end{bmatrix}
    ,
    \vec{v_2} =
    \begin{bmatrix}
    2 \\
    -3
    \end{bmatrix}
    [/tex]

    I found a widget on Wolfram Alpha that says the second eigenvector should be:
    [tex]
    \begin{bmatrix}
    2 \\
    3
    \end{bmatrix}
    [/tex]

    I am more inclined to believe wolfram alpha is correct, but can someone show me why?
     
  2. jcsd
  3. May 18, 2016 #2

    pasmith

    User Avatar
    Homework Helper

    How are we to determine whether, and if so how, you have made an error if you don't post your working?
     
  4. May 18, 2016 #3
    It was mostly by looking at the matrix and trying to make them the same and now that you say that when I do the algebra on it to post it, I see what I did wrong. I just tried to look at 3 and -2 to find the solution when I should have said 3x-2y=0 and 3x=2y then the vector becomes clear.
     
  5. May 18, 2016 #4

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    Neither of your ##v_1## or ##v_2## are eigenvectors. If ##A## is your matrix we have ##Av_1 = (8,12)^T##, which cannot be a multiple of ##v_1##: the first component of ##v_1## is larger than the second component, while the opposite is true for ##Av_1##. Similarly, ##v_2## cannot be an eigenvector of ##A## because the components of ##v_2## have opposite signs, while those of ##Av_2## have the same sign.

    Also: you did not show the eigenvalues.
     
  6. May 18, 2016 #5
    Eigens are 0,8. The problem was v2 I ommited a negative sign in the first vector on accident.
     
  7. May 18, 2016 #6

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    I do not understand what you are trying to say. What is the correct ##v_1##? What is the correct ##v_2##? Instead of trying to describe these in words, just show the actual numerical entries.
     
  8. May 18, 2016 #7
    [tex]
    \vec{v_1}=
    \begin{bmatrix}
    -2 \\
    1
    \end{bmatrix}
    ,
    \vec{v_2}=
    \begin{bmatrix}
    2 \\
    3
    \end{bmatrix}
    [/tex]

    Sory for putting so little work into this, I've got the answer and moved onto the next problem at this point.
     
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