ST and TS have the same eigenvals

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For linear operators S and T on a vector space V, it is established that the products ST and TS share the same eigenvalues. The discussion highlights the proof process, where if g is an eigenvalue and u is a nonzero eigenvector, then STu = gu leads to TS(Tu) = g(Tu). The challenge arises in ensuring that Tu remains nonzero, which is critical for maintaining the validity of the eigenvalue relationship. The zero eigenvalue case is identified as a special consideration that requires separate handling.

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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.
 
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If Tu is zero, what's STu?
 
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?
 

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