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January 17th, 2019, 12:59 PM   #1
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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!
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January 18th, 2019, 01:39 PM   #2
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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.
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January 18th, 2019, 07:40 PM   #3
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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.
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January 19th, 2019, 02:27 PM   #4
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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.
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