# Eigenvalues and eigenfunction

1. Jan 20, 2010

### 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

2. Jan 20, 2010

### Staff: Mentor

Isn't 5 the eigenvalue?

3. Jan 20, 2010

### glueball8

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?

4. Jan 20, 2010

### 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.

5. Jan 20, 2010

### vela

Staff Emeritus
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 $y'-\lambda y=0$ 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?

6. Jan 20, 2010

### glueball8

Ok cool. :)

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

Thanks

7. Jan 21, 2010

### Staff: Mentor

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