Although "fixation" carries an unintended negative connotation, why does it seem like random matrix theory is focused so intently on the eigenvalues of the matrices in question? Why is so little discussion given on definitions for algebra operations for these matrices?
Of course, it is difficult to provide thorough evidence of this without extensive discussion but I will briefly try:
- In this video titled "Random Matrices: Theory and Practice - Lecture 1", the lecturer states that random matrix theory is about saying "something" about the eigenvalues given joint distributions of matrix entries (around 3:00).
- In this paper whose title seems primarily concerned the products of random matrices a substantial portion of the content is dedicated to matrix eigenvalues.
- More than half of the wikipedia page on random matrices is concerned with distributions of their eigenvalues.
Does understanding the eigenvalues essentially "unlock" the ability to perform all other operations on these matrices? For example, can I find the expectation of the square of a random matrix without needing to find the distributions of its eigenvalues?