I believe I am very close to finishing this proof, but I cannot figure out the last part. If anybody could check my work and maybe give me a little hint, it would be greatly appreciated!
Let $V$ be a finite-dimensional vector space and $T \in \mathcal{L}(V)$, and let $\mathbf{u,v} \in V$ be eigenvectors of $T$.
Claim: If $\mathbf{u} + \mathbf{v}$ is an eigenvector of $T$, then $\mathbf{u}, \mathbf{v}$, and $\mathbf{u+v}$ all correspond to the same eigenvalue.
Proof (So far!): Suppose $T(\mathbf{u}) = \lambda_1 \mathbf{u}$ and $T(\mathbf{v}) = \lambda_2 \mathbf{v}$ with $\mathbf{u}, \mathbf{v} \neq \mathbf{0}$.
Now suppose $T(\mathbf{u+v}) = \lambda_3(\mathbf{u+v})$ with $\mathbf{u+v}\neq \mathbf{0}$.
Then, $$
T(\mathbf{u}) + T(\mathbf{v}) = \lambda_3 \mathbf{u} + \lambda_3 \mathbf{v}\\
\lambda_1\mathbf{u} + \lambda_2\mathbf{v} = \lambda_3 \mathbf{u} + \lambda_3\mathbf{v}\\
\lambda_1\mathbf{u} + \lambda_2\mathbf{v} – \lambda_3 \mathbf{u} -\lambda_3\mathbf{v} = \mathbf{0}\\
(\lambda_1 – \lambda_3) \mathbf{u} + (\lambda_2 – \lambda_3)\mathbf{v} = \mathbf{0}
$$
Now I know in order to show that $\lambda_1 = \lambda_2 = \lambda_3$, I must show that the only solution to the last line is the trivial one. This would imply that $\mathbf{u}$ and $\mathbf{v}$ are linearly independent which I am unconvinced of!
The only information I have to my advantage I haven't used yet is the fact that $\mathbf{u}, \mathbf{v}, \mathbf{u+v} \neq \mathbf{0}$. I really cannot see how this information can help me though.
Perhaps I am going about this wrong, but that's why I want to ask! Thanks for your help.
Best Answer
$$(\lambda_1-\lambda_3)\vec u+(\lambda_2-\lambda_3)\vec v=\vec 0$$ Case 1:
$\vec u$ and $\vec v$ are linearly independent. Then as you say, $\lambda_1-\lambda_3=\lambda_2-\lambda_3=0$, so done.
Case 2:
$\vec u$ and $\vec v$ are linearly dependent. Then $\vec v=k\vec u$ for a scalar $k$. Then $T(\vec v)=\lambda_2 \vec v$, but also $T(\vec v)=T(k\vec u)=kT(\vec u)=k\lambda_1\vec u=\lambda_1\vec v$. Since $\vec v$ is an eigenvector, it is non zero, and so $\lambda_1=\lambda_2$. Then you can go back to the previous equation $$\lambda_1 (\vec u+\vec v)=\lambda_3(\vec u+\vec v)$$and again using the fact that $\vec u+\vec v$ is an eigenvector so non-zero, it can be concluded that all the $\lambda_i$'s are equal.