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Suppose we have two real, symmetric and positive semidefinite matrices $A$ and $B$, and we know that they approximate each other well in the sense that

$$\| A - B \|_2 \le \epsilon,$$

where $\epsilon$ is positive and $\|\cdot\|_2$ is the $\ell_2$ operator norm. Is it possible to bound the difference between their leading eigenvalues, $|\lambda_1(A) - \lambda_1(B)|$, in terms of $\epsilon$?

I know it's possible to bound the angle between the leading eigenvectors of $A$ and $B$, but I have no idea how to prove a similar bound for eigenvalues.

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    $\begingroup$ The difference is bound by $\epsilon$ as long as both matrices are normal, see the Bauer-Fike theorem. $\endgroup$ – Conifold Jan 2 at 4:00
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We may assume that $\lambda_\max(A)\ge\lambda_\max(B)$. Then \begin{aligned} |\lambda_\max(A)-\lambda_\max(B)| &=\lambda_\max(A)-\lambda_\max(B)\\ &=\|A\|_2-\|B\|_2\\ &\le\|A-B\|_2 \le\epsilon. \end{aligned}

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