Differentiation Map of a Complex Transformation

[nautolian]

Homework Statement

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

Homework Equations

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.

 

[tiny-tim]

hi nautolian!

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

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

 

[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.

 

[Dick]

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.

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).

 

[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!

 

[Dick]

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!

Yes, that's it.