# Differentiation Map of a Complex Transformation

1. Nov 11, 2012

### nautolian

1. The problem statement, all variables and given/known data

Find the eigenvectors and eigenvalues of the differentiation
map C1(R) -> C1(R) from the vector space of differentiable functions
to itself.

2. Relevant equations

3. The attempt at a solution

Hi, I'm not entirely sure how to go about this, because would the differentiation map of this be [(1,0),(0,1)] since its from it to itself? Thanks for the help in advance.

2. Nov 11, 2012

### tiny-tim

hi nautolian!
an eigenvector is an element f of C1(R) such that Df is a scalar times f

3. Nov 12, 2012

### nautolian

High, sorry I'm still not really sure where to go with this. I mean I understand that Df=(lambda)f, but in the terms of C^1(R) does this mean that the derivative of the complex number a+bi is the same as the eigenvalue times the vector? Sorry, I'm still somewhat lost. Thanks for your help though.

4. Nov 12, 2012

### Dick

I don't think this has anything to do with complex numbers. C^1(R) usually just means differentiable real functions. You need to solve the differential equation f'(x)=λf(x).

5. Nov 14, 2012

### nautolian

okay, would that mean that there are infinite eigenvalues with associated iegenvectors equal to A*exp(lambda*x)? where A is a constant? Sorry I'm still unclear about this. Thanks for the help though!

6. Nov 14, 2012

### Dick

Yes, that's it.