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Questions where the notion matrix has an important or crucial role (for the latter, note the tag matrix-theory for potential use). Matrices appear in various parts of mathematics, and this tag is typically combined with other tags to make the general subject clear, such as an appropriate top-level tag ra.rings-and-algebras, co.combinatorics, etc. and other tags that might be applicable. There are also several more specialized tags concerning matrices.
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What is known about the distribution of eigenvectors of random matrices?
This is just a small update (much later!) on these interesting questions. On the 4th point of the OP, the distribution of the number of real eigenvalues can be found in my pre-print https://arxiv.org/ …