Bound eigenvalues $A$ and $B$ with 2-norm $\|A-B\|$

Question

Suppose $\| A-B\|_2 \leq\epsilon$, can we bound the difference in eigenvalues of $A$ and $B$ as well? For instance, would it hold that $|\lambda(A)-\lambda(B)|\leq\epsilon$ as well? Or alternatively, could we bound the singular values $|\sigma(A) - \sigma(B)|\leq\epsilon$?

Answer 1

Probably not unless you bound the matrix dimensions. Consider the zero matrix and the constant matrix $A(i, j) = \varepsilon$ which has one eigenvalue $n\times \varepsilon$ for $A\in \mathbb{R}^{n\times n}$.

Note also that in order to make the question more precise, you should specify what exactly you mean by $|\lambda(A) - \lambda(B)|$ as these are sets (of possibly different cardinality).