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Finding eigenvectors of a simple 2x2 matrix

  1. Dec 2, 2009 #1
    1. The problem statement, all variables and given/known data

    The matrix is:
    |1 2|
    |3 4|


    3. The attempt at a solution

    I've worked out the eigenvalues to be [tex]\stackrel{\underline{5\pm\sqrt{33}}}{2}[/tex]

    But when I plug the first eigenvalue back in I get:

    |1 - [tex]\stackrel{\underline{5+\sqrt{33}}}{2}[/tex]........................2 |
    |3........................4 - [tex]\stackrel{\underline{5+\sqrt{33}}}{2}[/tex] |

    Multiplied by (x,y) = (0,0)

    [Sorry I couldn't fit that on the same line as the matrix.]


    Which in turn gives two equations:

    x*(1-[tex]\stackrel{\underline{5+\sqrt{33}}}{2}[/tex]) + 2y = 0

    and


    3x + y*(4-[tex]\stackrel{\underline{5+\sqrt{33}}}{2}[/tex]) = 0


    Which I can't work out how to solve for the eigenvector.

    Is the only solution x=y=0?
     
  2. jcsd
  3. Dec 2, 2009 #2

    Dick

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    Science Advisor
    Homework Helper

    You solve them like you solve any two linear equations. Remember if (x,y) is an eigenvector then a*(x,y) is also an eigenvector. So you are going to have an undetermined parameter which you may as well may make x. Put say, x=1 and solve for y. You should get the same value of y from both equations, since one equation is really a multiple of the other.
     
  4. Dec 2, 2009 #3
    Thanks for the reply!

    Thats exactly what I've been trying to do but X and/or Y always end up canceling and I just get some useless number.
     
  5. Dec 2, 2009 #4

    Dick

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    Science Advisor
    Homework Helper

    Are you trying to eliminate a variable from the two equations and solve for the other? That won't work. If you eliminate x then the resulting equation is 0*y=0. Is that your problem? Like I said before, that's happening because the two equations are really the same. They have an infinite number of solutions. Put x equal to any constant (like x=1) and then solve for y.
     
  6. Dec 2, 2009 #5
    AH!

    Thank you!
     
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