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Let $Q=(q_{j,k})_{1\le j,k\le N}$ be a (Hermitian) $N\times N$ matrix with complex-valued entries. The matrix $Q$ is given numerically and the absolute error on each entry is bounded above by a (small) positive number $\eta$. We deal actually with $\tilde Q=(\tilde q_{j,k})_{1\le j,k\le N}$, with $\vert\tilde q_{j,k}- q_{j,k}\vert\le \eta$.

I would like to find an upper bound for the absolute error on the eigenvalues of $Q$: we find numerically $\tilde\lambda_1,\dots, \tilde\lambda_N$ and if $\lambda_1,\dots, \lambda_N$ are the eigenvalues of $Q$ (which are real-valued), how can I bound from above the quantity $\vert\tilde\lambda_j-\lambda_j\vert$?

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    $\begingroup$ Obviously, you can bound the differences by the norm of the error matrix ($N\eta$ in the worst case scenario, $C\sqrt N\eta$ on average) and in general you can hardly do any better. $\endgroup$
    – fedja
    Commented Jun 20, 2018 at 13:50
  • $\begingroup$ To get the tightest possible bound in any particular case, and for a particular eigenvalue, you can apply global numerical optimization to the non-convex problem of maximizing the absolute value of eigenvalue difference subject to matrix being hermitian and within $\eta$ in magnitude of Q in each element.of the upper triangle. $\endgroup$
    – Mark L. Stone
    Commented Jun 20, 2018 at 14:22
  • $\begingroup$ Terry Tao has an old post on this terrytao.wordpress.com/2008/10/28/when-are-eigenvalues-stable $\endgroup$
    – j.c.
    Commented Jun 20, 2018 at 14:25

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