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: Eigenvalues and eigenfunction

  1. Jan 20, 2010 #1
    1. The problem statement, all variables and given/known data
    T(f(x)) = 5 f(x)
    T is defined on C. Find all real eigenvalues and real eigenfunction. V:R -> R

    2. Relevant equations
    Not sure.


    3. The attempt at a solution
    No, clue. I can find eigenvalues for matrices, that's not a problem. I'm having problem that its a T(function) = something function, how do I solve a problem like this in general?

    Any hints?

    Thanks
     
  2. jcsd
  3. Jan 20, 2010 #2

    Mark44

    Staff: Mentor

    Isn't 5 the eigenvalue?
     
  4. Jan 20, 2010 #3
    :redface: Bad example. So the eigenfunction is any function?

    Uh how to do theses in general? Can you make up something more complicated and explain how to do it?

    Or a link?
     
  5. Jan 20, 2010 #4

    HallsofIvy

    User Avatar
    Science Advisor

    How about using the definition of "eigenvalue": If T is a linear transformation that maps functions into functions, then [itex]\alpha[/itex] is an "eigenvalue" and a non-zero function, f(x), is an eigenvector if and only if [itex]Tf(x)= \alpha f(x)[/itex]. If you are told that Tf(x)= 5f(x) for all f, then, yes, 5 is the only eigenvalue and every function in the space is an eigenvector. T just "multiplies by 5" and is exactly the same as a diagonal matrix having all "5" on its diagonal.
     
  6. Jan 20, 2010 #5

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    If all T does is multiply a function by 5, then yes.

    Usually, you solve a differential equation. Equations like the Legendre's differential equation, the Bessel's differential equation, and the Schrodinger equation are all of this form.

    The differential equation [itex]y'-\lambda y=0[/itex] is a simple example. You could write it as

    [tex]D(y) = \lambda y[/tex]

    where D is the derivative operator. The solution to this equation [itex]y=e^{\lambda x}[/itex] is an eigenfunction of D.

    Did you have a specific type of problem in mind?
     
  7. Jan 20, 2010 #6
    Ok cool. :)

    How about T(f(x)) = 4f(-x) + f'(x) + 6f(6)? (Might not be do able) Or something like that?

    Thanks
     
  8. Jan 21, 2010 #7

    Mark44

    Staff: Mentor

    Give us an actual problem, not something you just made up.
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook