Eigenvectors and Eigenvalues in relation to a double transformation

Dec 2018
4
0
Tel Aviv
Hi guys,

I have the following question: I have been thinking about this for a while now, but I simply cannot work it out...

Let V be a linear space of n dimensions over R, and let S,T:V->V be linear transformations.

True or False?

1. If v is an eigenvector of S and of T, then v is also an eigenvector of S + T.
2. If λ_1 is an eigenvalue of S and λ_2 is an eigenvalue of T, then λ_1 + λ_2 is a eigenvalue of S + T.

I am not just looking for the right answer, but also for the reasoning behind it...

Thank you!
 

mathman

Forum Staff
May 2007
6,913
762
1) (S+T)(v)=S(v)+T(v)=cv+bv=(c+b)v.
2) Yes if the eigenvector is the same for both (like in 1) above). Otherwise possibly by coincidence only.
 

SDK

Sep 2016
743
497
USA
2) Yes if the eigenvector is the same for both (like in 1) above). Otherwise possibly by coincidence only.
Even in this case it can still easily fail. Take both matrices to be the identity for example.
 

mathman

Forum Staff
May 2007
6,913
762
Even in this case it can still easily fail. Take both matrices to be the identity for example.
Why? If both are identity, then every vector is an eigenvector, with an eigenvalue of 2 for the sum.