Say $B$ is a real symmetric matrix of dimension $n$ and $A$ is another real symmetric matrix of the same dimension.

• What needs to be the bounds on the norm of $B$ to ensure that $\lambda_{max}(A+B) - \lambda_{max}(A) = O(\frac{1}{n} )$ ?

Intuitively it seems that one needs the $l_{\infty}$ norm of $B$ to be low compared to $A$. But I am looking for more quantitative statements to hit that $O(\frac{1}{n})$ thing.