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?
