How can you find the inverse of a polynomial with a nilpotent operator?

[johnson123]

Homework Statement

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}

 

[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.

 

[johnson123]

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

is invertible?

 

[Dick]

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

is invertible?

Good point. I was solving the wrong problem.

 

[Hurkyl]

I also tried a_{0}^{-1}T^{k-1}
but this gives me T^{k-1}

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

 

[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?