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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
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 Originally Posted by mathman 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. January 19th, 2019, 02:27 PM   #4
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 Originally Posted by SDK 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. Tags double, eigenvalue, eigenvalues, eigenvector, eigenvectors, linear transformation, relation, transformation Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post integration Elementary Math 1 March 27th, 2018 04:20 AM calypso Linear Algebra 4 June 21st, 2016 08:46 AM hk4491 Linear Algebra 3 March 4th, 2014 01:03 PM international Linear Algebra 2 July 10th, 2013 04:01 PM baz Linear Algebra 1 April 11th, 2013 10:28 AM

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