Eigenvalues of the sum of a symmetric matrix $A$ and a hermitian matrix $B$

Question

If I have two matrices: $A \in \mathbb C$ symmetric, and $B \in \mathbb C$ hermitian.

Question 1: If the spectral radius of $A$ is much larger than the spectral radius of $B$, can I say that the eigenvalues ​​of $(A + B)$ are close to the eigenvalues ​​of $A$? That is, for each eigenvalue $ A $, is there an eigenvalue of $ A + B $ of approximate value?

Question 2: If the infinite norm of $A$ is much larger than the infinite norm of $B$. Can I say that the eigenvalues of $A+B$ are close to the eigenvalues of $A$?

Answer 1

What do you mean by "similar" for eigenvalues?

EDITED: I assume your $A$ is a real symmetric matrix. Let $\lambda_i(A)$ and $\lambda_i(A+B)$ be the eigenvalues of those matrices, each sorted in increasing order, $\lambda_{min}(B)$ and $\lambda_{max}(B)$ the minimum and maximum eigenvalues of $B$. Then $$\lambda_i(A) + \lambda_{min}(B) \le \lambda_i(A+B) \le \lambda_i(A) + \lambda_{max}(B)$$
This follows from the minimax principle.