# Nilpotent Operator

1. Mar 27, 2008

### johnson123

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

If T is a nilpotent transformation from V -> V, V - finite dimensional vector space.

show that $$a_{0}+a_{1}T+\cdots+a_{k}T^{k}$$ is invertible. $$a_{0}$$ nonzero.

Im having trouble finding the inverse, I know for 1+T+$$\cdots$$+T$$^{m-1}$$

the inverse is (1-T),where T$$^{m}$$=0. I also tried $$a_{0}^{-1}T^{m-1}$$
but this gives me $$T^{m-1}$$

Last edited: Mar 27, 2008
2. Mar 27, 2008

### Dick

Write that as -a0*I=a_1*T+a_2*T^2+...+a_k*T^k. Is that enough of a hint? If not notice the right side has a common factor.

3. Mar 27, 2008

### johnson123

I see that it shows T to be invertible but how does it show that the original polynomial

is invertible?

4. Mar 27, 2008

### Dick

Good point. I was solving the wrong problem.

5. Mar 27, 2008

### Hurkyl

Staff Emeritus
Okay, so you figured to annihilate all of the terms of your polynomial except the T^{k-1} term. What else can you annihilate?

6. Mar 27, 2008

### Dick

I think you are going to have a hard time expressing an answer in closed form in terms of the a_i's. But think of it this way. If P is your polynomial, then P=a0+Q where Q is the rest of the terms. Can you show Q is nilpotent? Does that suggest how to invert a0+Q?