Homework Help: ST and TS have the same eigenvals

1. Nov 2, 2006

balletomane

Hi.

I need to prove that for S,T linear operators on V. ST and TS have the same eigenvalues. I've gotten as far as (say g is the eigenvalue and u is a nonzero vector): STu=gu so TS(Tu)=g(Tu). So TS has eigenvalue g corresponding to eigenvector Tu. But I don't know how to guarantee that Tu is nonzero. (Or how to resolve the possibility that it is zero)

Thanks for any help.

2. Nov 2, 2006

Office_Shredder

Staff Emeritus
If Tu is zero, what's STu?

3. Nov 2, 2006

balletomane

If Tu=0

STu=0=gu, but u is nonzero, so g=0?

So does zero as an eigenvalue become a special case? Then would I do the nonzero eigenvalue case separately?