My question pertains eigenvectors of matrices with somewhat evenly distributed entries.
Let $M$ be an $N \times N$ matrix with complex entries (think of $N$ as a large integer). You can assume that $M$ is unitary if you like (it is in the particular question that I have in mind but my question makes sense regardless).
Now I'm going to assume that the entries of my matrix are all somewhat small (let's say less than $C N^{-2}$ for a constant $C> 0$ if $M$ is unitary, but again feel free to change this hypothesis if it makes for some interesting result).
Question : can I infer interesting things about the eigenvectors of $M$ ? For instance, can one say that the entropy of an eigenvector of $M$ is large in some quantitative way ? (what I call the entropy of $(z_1, \cdots, z_n) \in \mathbb{C}^N$ is $\sum_1^N{- |z_i| \log |z_i|}$, I don't know if it is standard terminology).
I am interested in non-probabilistic results/literature, I understand that there are a lot of interesting things to say about the distribution of eigenvectors when coefficients are taken at random.
(Remark: You need to assume something on the matrix (unitary, determinant $1$, ...) to make the hypothesis 'the entries are small' relevant, but as I'm interested in the relationship between entries and eigenvectors I don't want to be too specific.)
I appreciate that it is a relatively basic question and would be grateful to just be pointed to relevant literature. Many thanks :)
