Upper bound on the spectral radius of summation of two matrices one symmetric one diagonal

Question

I am looking for an upper bound for the largest absolute value of eigenvalues of the summation of two matrices $\underset{i}\max\{|\lambda_i(A+K)|\}$ where $A$ is symmetric with real nonnegative eigenvalues and $K$ is a diagonal matrix.

Any idea is appreciated.

Thanks.

Answer 1

[EDITED] If $M$ and $m$ are the greatest and least diagonal elements of $K$ respectively, $$\lambda_{min}(A)+m = \lambda_{min}(A+mI) \le \lambda_{min}(A+K) \le \lambda_{max}(A+K) \le \lambda_{max}(A+MI) = \lambda_{max}(A)+M$$ Thus $|\lambda_i(A+K)| \le \max(|\lambda_{min}(A)+m|, |\lambda_{max}(A)+M|)$.