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January 17th, 2019, 12:59 PM  #1 
Newbie Joined: Dec 2018 From: Tel Aviv Posts: 4 Thanks: 0  Eigenvectors and Eigenvalues in relation to a double transformation
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! 
January 18th, 2019, 01:39 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,823 Thanks: 723 
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. 
January 18th, 2019, 07:40 PM  #3 
Senior Member Joined: Sep 2016 From: USA Posts: 646 Thanks: 411 Math Focus: Dynamical systems, analytic function theory, numerics  
January 19th, 2019, 02:27 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,823 Thanks: 723  

Tags 
double, eigenvalue, eigenvalues, eigenvector, eigenvectors, linear transformation, relation, transformation 
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