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Homework Help: Eigenvector Basics

  1. Nov 24, 2008 #1
    1. The problem statement, all variables and given/known data

    Find the eigenvectors of the following matrix:

    [tex]
    \[ \left( \begin{array}{ccc}
    1 & 1 \\
    4 & -2 \end{array} \right)\]
    [/tex]

    2. Relevant equations

    N/A

    3. The attempt at a solution

    I already know how to find the solutions. They are {1 1} and {-1 4}. My question is this: could a solution also be {1 -4} (switching the minus sign)? I don't think it is a solution but I don't see why not...

    Any help would be great. Thanks!
     
  2. jcsd
  3. Nov 24, 2008 #2
    Have you tried [tex]\begin{pmatrix}1 \\ -4\end{pmatrix}[/tex] ? All it takes to be a solution is solving the eigenvalue problem for some eigenvalue [tex]\lambda[/tex].

    Tom
     
  4. Nov 24, 2008 #3
    your sol'n times any c in R is fine
     
  5. Nov 24, 2008 #4
    Oh I guess that's obvious huh. So if c=-1, I'm good.

    Okay thanks!
     
  6. Nov 24, 2008 #5
    Exactly.
     
  7. Nov 24, 2008 #6

    HallsofIvy

    User Avatar
    Science Advisor

    One important thing you should have learned (or maybe this exercise is in preparation) the set of all eigenvectors (corresponding to a given eigenvalue) forms a subspace: it is closed under both vector addition and scalar multiplication.
     
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