# Eigenvectors and Eigenvalues in relation to a double transformation

#### pregunto

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 Î»_1 is an eigenvalue of S and Î»_2 is an eigenvalue of T, then Î»_1 + Î»_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
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

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
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.