# Homework Help: Jordan canonical form 1

1. May 15, 2010

### tinynerdi

1. The problem statement, all variables and given/known data
Let T:V->W be a linear transformation. Prove that if V=W (So that T is linear operator on V) and λ is an eigenvalue on T, then for any positive integer K
N((T-λI)^k) = N((λI-T)^k)

2. Relevant equations
T(-v) = -T(v)
N(T) = {v in V: T(v)=0} in V hence T(v) = 0 for all v in V.

3. The attempt at a solution
we know that (T-λI)^k(-v) = -(T-λI)^k(v) = (λI-T)^k(v). So when (T-λI)^k(v) = 0 so does (λI-T)^k(v). Hence N((T-λI)^k) = N((λI-T)^k)

2. May 16, 2010

### Landau

You mean Tv=0 for all V in N.
You mean (T-λI)^k(v) = 0 if and only if (λI-T)^k(v) = 0.

3. May 16, 2010

### tinynerdi

Yeah, that is what I am trying to prove.

4. May 16, 2010

### Landau

It looks correct. Basically, you're proving that for any linear map T, T and -T have the same kernel. This is true because Tv=0 iff -Tv=0.

5. May 16, 2010

### tinynerdi

Can we just state that because Tv=0 iff and -Tv = 0 therefore N((T-λI)^k) = N((λI-T)^k) or do we have to prove that Tv=0 iff and -Tv = 0?

6. May 17, 2010

### Landau

Well, there is not much to prove. The only thing you need is that -0=0.