Norm bounds on spectral variation and eigenvalue variation

The Hermitian case is more like a state-of-art answer. A good review of results in given in [Holbrook].

$\nu(A,B)\leq\|A-B\|$ for the operator norm. This is a direct consequence from Weyl's inequality.

This problem about spectral variation bound is fully discussed in [Bhatia] Chap 3&4(with a supplement in Chap7&8 if you got the 2006 ed.)

“The spectral variation problem for the class of Hermitian matrices has been completely solved in the following sense. For any two Hermitian matrices a tight upper bound for the distance between their eigenvalues is known. Such bounds are known when the distance is measured in any unitarily-invariant norm.”[Bhatia]p.34

Later development including marjorization inequalities used in controlling covariance matrices in statistics as motivation [Marshall&Olkin], as described in[Bhatia] 3.9

(This is quite clear once you know the reference, probably that is why it gets downvotes.)

Reference

[Holbrook]Holbrook, John A. "Spectral variation of normal matrices." Linear algebra and its applications 174 (1992): 131-144.

[Bhatia]Bhatia, Rajendra. Perturbation bounds for matrix eigenvalues. Society for Industrial and Applied Mathematics, 2007.

[Marshall&Olkin]Marshall, Albert W., Ingram Olkin, and Barry C. Arnold. Inequalities: theory of majorization and its applications. Vol. 143. New York: Academic press, 1979.