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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 (which?) 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.

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  • $\begingroup$ One cheap bound (maybe not what you're looking for): From Weyl's inequality the difference is bounded by $||B||_2$. $\endgroup$
    – Kevin P. Costello
    Commented Aug 21, 2015 at 19:20
  • $\begingroup$ @KevinP.Costello I guess you mean is this : One knows that $\lambda_{max}(A+B) - \lambda_{max}(A) \leq \lambda_{max}(B)$ and hence if $\lambda_{max}(B) = O(\frac{1}{n})$ then my needed property follows. Right? $\endgroup$
    – user6818
    Commented Aug 21, 2015 at 19:24
  • $\begingroup$ Pretty much, though you need a bound on both $\lambda_{max}$ and $\lambda_{min}$ if you want a $2$-sided bound on the max eigenvalue. $\endgroup$
    – Kevin P. Costello
    Commented Aug 21, 2015 at 20:28
  • $\begingroup$ What do you mean by a 2-sided bound? The $O$ expression I wrote is not 2-sided, I guess. $\endgroup$
    – user6818
    Commented Aug 21, 2015 at 23:06
  • $\begingroup$ Two-sided meaning $\lambda_{max}(A+B)$ is close to $\lambda_{max}(A)$ (as opposed to just saying $\lambda_{max}(A+B)$ is not much larger than $\lambda_{max}(A)$). $\endgroup$
    – Kevin P. Costello
    Commented Aug 21, 2015 at 23:16

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