# Find the eigenvalues of this endomorphism of R[X]

1. Feb 15, 2010

### penguin007

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

f is an endomorphism of Rn[X]
f(P)(X)=((aX+b)P)'

eigenvalues of f?

2. Relevant equations

(a,b)<>(0,0)

3. The attempt at a solution

If a=0, then f(P)=bP', and only P=constant is solution

if a<>0, then I put Q=(ax+b)P, f(P)=cP is equivalent to (ax+b)Q'=Q (E)

I solved (E) and found Q(X)=(aX+b)^c but then if I say P(X)=(aX+b)^(c-1), I can't find c...

2. Feb 15, 2010

### vela

Staff Emeritus
Re: eigenvalues

Is f(P)=(aX+b)P' or [(aX+b)P]'? You wrote it both ways.

3. Feb 16, 2010

### penguin007

Re: eigenvalues

f(P)=[(aX+b)P]'

4. Feb 16, 2010

### vela

Staff Emeritus
Re: eigenvalues

Sorry, I misread your initial post. I see what you did now. Your solution for Q(X) should be $Q(X)=(aX+b)^{c/a}$ so $P(X)=(aX+b)^{c/a-1}$. Do you see now what values c can be?

5. Feb 16, 2010

### penguin007

Re: eigenvalues

Thanks a lot vela, I mistook when I solved (E)... Now I can see the values for c (s.t c/a-1 is integer and therefore P is a polynom)...