1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Eigenvector help

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

    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
    Last edited by a moderator: May 2, 2017
  2. jcsd
  3. Jan 24, 2007 #2
    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.

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


    User Avatar
    Science Advisor

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook