Eigenvalues and eigenfunction

[glueball8]

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

[Mark44]

Isn't 5 the eigenvalue?

[glueball8]

Isn't 5 the eigenvalue?

: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?

[HallsofIvy]

How about using the definition of "eigenvalue": If T is a linear transformation that maps functions into functions, then \alpha is an "eigenvalue" and a non-zero function, f(x), is an eigenvector if and only if Tf(x)= \alpha f(x). 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.

[vela]

:redface: Bad example. So the eigenfunction is any function?
If all T does is multiply a function by 5, then yes.

Uh how to do these in general? Can you make up something more complicated and explain how to do it?
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 y'-\lambda y=0 is a simple example. You could write it as

D(y) = \lambda y

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

Did you have a specific type of problem in mind?

[glueball8]

Ok cool. :)

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

Thanks

[glueball8]

bump

[Mark44]

Give us an actual problem, not something you just made up.