## Jordan basis

Hi Sir

I was looking at 2006 qs 4c) which asks me to find the Jordan Canonical Form of D, where

V= {a0 + a1x + a2x^2 : a0, a1, a2 in C}

and D: V–>V defined as D(f) = d^2f/dx^2 – 4f

I have been looking at Homework 4 and 5 to help me with this but both examples are without the 4f.

I want to find ch(x). If we have polys of deg <= 2 we need to differentiate 3 times to get 0. So D^2 will give me 0. But D^2(a0 + a1x + a2x^2) = -4(a0 + a1x + ax^2)

(I hope I am correct in thinking that f= a0 + .. regardless of the power of D) So for this to be 0 we need a0 = 0 and x=0. From here I am not sure how to go about finding ch(x) and m(x) (I would guess that ch(x) = x^3 and m(x)= x^2 but without much confidence or knowledge), would you be able to help me on this?

Thank you

Kind Regards

Reply:

There are many mistaken statements above. For example, D is not just differentiation, so no power of it needs to be zero.

Let me suggest one easy way to do this problem. Presumably, you know how to find the Jordan canonical form and Jordan basis for a 3 by 3 matrix. Can you turn this problem into one about such a matrix? However, remember that the eventual Jordan basis needs to be expressed in terms of elements of V.

Ask again if you run into trouble.

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