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In my new paper, one of the main theorems gives an upper bound for the spectral distance of a general real symmetric matrix to diagonal matrices:

Theorem 3. ‎Let $A=[a_{ij}]$ be a real symmetric matrix of order $n>1$‎. ‎Then‎ ‎$$\min_{D\in \mathcal{D}_n} \Vert{A-D}\Vert_{\infty} \;\leq\; \frac{n}{2}\max_{i\neq j} |a_{ij}|,$$ ‎and the bound is sharp for all $n>1$‎.

($\mathcal{D}_n$ stands for the space of diagonal matrices of order $n$ and $\Vert A\Vert_\infty$ denotes the "spectral" norm of the matrix $A$, which is the maximum singular value.)

It seems to me that this is a new and nontrivial bound for the distance to the nearest diagonal matrix, and should have some interesting applications or consequences in other areas of mathematics, specially algebraic combinatorics or quantum coherence. For the second, I also read a bunch of papers on coherence measure, without obtaining any remarkable relation between the above theorem and works there.

Does anyone have ideas or suggestions in this respect?

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    $\begingroup$ Statistics, and in particular diagonal approximations to covariance matrices. One might try to bound how much worse is a model based on the best diagonal matrix is than the full matrix model. But that's only if you consider statistics to be an area of mathematics ;). $\endgroup$
    – Oxonon
    Oct 25, 2023 at 18:23
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    $\begingroup$ The operator norm, since we would usually care about worst-case performance. $\endgroup$
    – Oxonon
    Oct 25, 2023 at 22:02
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    $\begingroup$ The distance between a matrix and the closest diagonal matrix (usually measured in the trace norm in this context) comes up naturally in quantum information theory when trying to quantify "how coherent" a quantum state is. See arXiv:1551.01854, for example. $\endgroup$ Oct 26, 2023 at 11:37
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    $\begingroup$ related: mathoverflow.net/q/349044/11260 $\endgroup$ Oct 28, 2023 at 9:38
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    $\begingroup$ @Mostafa Just to clarify, let me note that it is always necessary to mention that you are the author of a paper you mention in your messages, by the rules in mathoverflow.net/help/promotion . Thanks for complying! $\endgroup$ Oct 30, 2023 at 14:17

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