Hermitian matrices and their eigenvalues

Question

Let $C=A+B$ where $A$ and $B$ are two hermitian matrices can I prove that $\lambda_{i,C}=\lambda_{i,A}+\lambda_{i,B}$ iff $x_{i,A}=x_{i,B}$? Where $x_i$ is the eigenvector related to eigenvalue $\lambda_i$.

Answer 1

Let $A=\begin{bmatrix} 1 & 0\\ 0 & 2\end{bmatrix}$ and $B=\begin{bmatrix} 0 & 0\\ 0 & 0\end{bmatrix}$.

Clearly the first equality always holds, however $\begin{bmatrix} 1 \\ 1\end{bmatrix}$ is an eigenvector of $B$, but not of $A$.