Eigenvalue of Sum of Eigenvectors

[TranscendArcu]

Homework Statement

The Attempt at a Solution

So, first I wrote,

$T(X) = λ_1 X, T(Y) = λ_2 Y$

If $λ_1 = λ_2$:

$T(X+Y) = T(X) + T(Y) = λ_1 X + λ_2 Y = λ_1 (X+Y)$,

so this does indeed seem to be an eigenvector. But I'm less convinced for the case $λ_1 ≠ λ_2$. Again, I get the transformation down to the form:

$λ_1 X + λ_2 Y$

But if eigenvalues are necessarily one constant, then I don't see how I pull constants out as above. How do I go about showing this?

 

[HallsofIvy]

Did you read the problem carefully? It doesn't ask you to show that this is true, it asks you to determine whether or not it is true and the show that.

Okay, if you have doubts about this being true, how about trying to show its not true? If it is not, you should be able to show a counterexample.
Obviously
$$\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$$
has 1 and 2 as eigenvalues. What are corresponding eigenvalues? What do you get if you multiply that matrix by their sum?

 

[Mark44]

What are corresponding eigenvalues?

I'm pretty sure HallsOfIvy meant "corresponding eigenvectors".

 

[HallsofIvy]

Yes, thanks.