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Eigenvector help

  1. Jan 23, 2007 #1
    http://orion.math.iastate.edu/vika/cal3_files/lec33267.pdf

    i searched eigenvectors on google and this showed up. here are some problems i need further explaining for example 3 and 4.

    3. where do they get e1+e2= 0 equation from? then where did they get e= (1,-1)

    4. where do they get the e1+2e2+e3= 0 eq from?

    any help would really be appreciated...i understand how to get the eigenvalues but i have no clue what to do to get the vectors
     
  2. jcsd
  3. Jan 24, 2007 #2
    3.
    Write out the vector [tex] \vec{e} [/tex] explicitly, i.e. [tex] (e_1,e_2) [/tex], then substitude into [tex] (A-3E) \vec{e} = 0 [/tex], you will see the result immediately.

    4.
    Same as above.........
     
  4. Jan 24, 2007 #3

    HallsofIvy

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    The first matrix is
    [tex]\left(\begin{array}{cc}-2 & 1 \\ -1 & 4\end{array}\right)[/tex]
    which, by solving the "characteristic equation" is determined to have the single eigenvalue -3.

    The definition of "eigenvalue" is that there exist a non-trivial (i.e. non-zero) vector v such that [itex]Ax= \lambda x[/itex]. Saying that -3 is an eigenvalue means that there is a non-zero vector
    [tex]v= \left(\begin{array}{c}x \\ y\end{array}\right)[/tex]
    such that
    [tex]\left(\begin{array}{cc}-2 & 1 \\ -1 & 4\end{array}\right)\left(\begin{array}{c}x \\ y\end{array}\right)= \left(\begin{array}{c}-2x+ y \\ -x+ 4y\end{array}\right)= \left(\begin{array}{c}-3x \\ -3y\end{array}\right)[/tex].

    The top row says -2x+ y= -3x and the second -x+ 4y= -3y. Normally, two equations in two variables would have a unique solution (and obviously x=0, y= 0 is a solution) but here the equations are not "independent" (precisely because -3 is an eigenvalue). It's easy to see that both equations reduce to y= -x. That is the same as x+y= 0 (your book uses e1+e2= 0 but it is the same thing). Any vector of the form (a, -a)= a(1,-1) is an eigenvector corresponding to eigenvalue -3.
     
  5. Jan 25, 2007 #4
    thanks for the help, hopefully it will make sense with time
     
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