# another question about eigenvalues and linear operators

• Jul 29th 2009, 03:34 AM
vonflex1
another question about eigenvalues and linear operators
hi, i have another question, last one - so i hope you won't mind:

i need to prove that:

S and T are two linear operators that can switch sides (ST = TS) over a vector space V.
lambda is a eigenvalue of S. the group W is defined as Ker ((lambda)I - S).
prove that W is contained in T(W).

thanks in advance to anyone who can help.
• Jul 29th 2009, 03:55 PM
NonCommAlg
Quote:

Originally Posted by vonflex1
hi, i have another question, last one - so i hope you won't mind:

i need to prove that:

S and T are two linear operators that can switch sides (ST = TS) over a vector space V.
lambda is a eigenvalue of S. the group W is defined as Ker ((lambda)I - S).
prove that W is contained in T(W).

thanks in advance to anyone who can help.

are you sure it's not the other way round, i.e. $T(W) \subseteq W$? that's a trivial result: if $w \in W,$ then $S(w)=\lambda w$ and so $ST(w)=TS(w)=\lambda T(w),$ which means $T(w) \in W.$ so $T(W) \subseteq W.$