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Find the eigenvalues of this endomorphism of R[X]

  1. Feb 15, 2010 #1
    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. jcsd
  3. Feb 15, 2010 #2

    vela

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    Re: eigenvalues

    Is f(P)=(aX+b)P' or [(aX+b)P]'? You wrote it both ways.
     
  4. Feb 16, 2010 #3
    Re: eigenvalues

    f(P)=[(aX+b)P]'
     
  5. Feb 16, 2010 #4

    vela

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    Re: eigenvalues

    Sorry, I misread your initial post. I see what you did now. Your solution for Q(X) should be [itex]Q(X)=(aX+b)^{c/a}[/itex] so [itex]P(X)=(aX+b)^{c/a-1}[/itex]. Do you see now what values c can be?
     
  6. Feb 16, 2010 #5
    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)...
     
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