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: A question about a new way to find eigenvectors that i noticed

  1. Mar 2, 2008 #1
    i got this question in which we are given the matrix T
    and we need to find the eigenvalues and the independent spaces (i dont know what is independent space) of T^2 +2*T

    the problem is that he started to solve the question as i would have solved it
    but then he puts a big X on it and does something else
    i cant understand it??(and he gets all the point for it)

    it looks as if he skips the finding the roots of polinomial step

    Last edited: Mar 2, 2008
  2. jcsd
  3. Mar 2, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    The solver appears to have realized that he didn't have to compute the eigenvalues of T^2+2*T since he already knew that the eigenvalues of T were +1 and -1, apparently from a previous problem. This let him immediately conclude the eigenvalues of T^2+2*T are 3 and -1. Once he knew the eigenvalues he substituted them in for lambda and seems to have read off the eigenvectors more or less by inspection. It's not a new way of computing eigenvalues.
  4. Mar 2, 2008 #3


    User Avatar
    Science Advisor

    If [itex]\lambda[/itex] is an eigenvalue of T, with eigenvector v, then Tv= [itex]\lambda[/itex]v. From that, [itex](T^2+ 2T)v= T(T(v))+ 2T(v)= T(\lambda v)- 2\lambda= \lambda T(v)- 2\lambda= \lambda(\lambda v)- 2\lambda v= (\lambda^2- 2\lambda) v[/itex].

    In other words if [itex]\lambda[/itex] is an eigenvalue of T with eigenvector v, then [itex]\lambda^2- 2\lambda[/itex] is an eigenvalue of T2- 2T with eigenvector v.

    It is easier to find the eigenvalues of T and then use that formula than to find the eigenvalues of T2- 2T directly.
    Last edited by a moderator: Mar 4, 2008
  5. Mar 4, 2008 #4
    ok i understood how you got the formula from the T^2 + 2T epression

    what now??
    how do i mix up the eigenvalues of T with this formula in order to get the new values
  6. Mar 4, 2008 #5


    User Avatar
    Science Advisor

    What do you mean by "mix up the eigenvalues of T" and what "new values" are you talking about?

    If you mean "How do I go from the eigenvalues of T to the eigenvalues of T2- 2T?", that's exactly what I told you before.:
  7. Mar 4, 2008 #6
    correct me if i am wrong

    x-eigenvalue of T
    y-eigen value of the expression

    so if x=-1 then for that "old" eigen value we get y=3
    and we do that proccess for every eigenvalue
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook