No one answers my questions, but I figure I might as well try.
In solving a problem I've stumbled across something that I'd like to verify. Suppose and is a symmetric matrix. Then, for any function the following holds:
The idea behind this is to apply the spectral theorem to write and define . Then, and .
From there, argue
(The are the eigenvalues of if that isn't clear). I guess the main thing I need to get this off the ground is that are identically distributed for all i. This seems true since I think is identically distributed for all i. This justifies the steps where I do stuff with the expected values.