1
$\begingroup$

I understand that there are invariant Haar measure for eigenvalues of unitary matrix. I further understand that absolute value of eigenvalues of unitary matrix is 1. But, I could not find any analysis on the phase of the eigenvalues. I am interested to attribute a probability measure to the phase of eigenvalues of unitary matrix. Is there any work in this line?

Improve this question
$\endgroup$

2 Answers 2

Reset to default
4
$\begingroup$

This paper by Diaconis should be what you're looking for.

In particular, it is known that the eigenvalues are not randomly spread around the unit circle as one's intuition would suggest, but much more evenly spaced apart. See the first of the "five surprising facts" in Section 4 of that paper.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Well, they are randomly spread, but they're not independent. $\endgroup$
    – Marcel
    Commented Mar 16, 2016 at 10:48
0
$\begingroup$

Eigenvalues of random unitary matrices are very well understood. This model of correlated random variables was introduced by Freeman Dyson in the 60's. We know their joint probability distribution function, which is proportional to $\prod_{jk}|z_j-z_k|^2$, we know their correlation functions, we know the moments of the characteristic polynomials, we know how their truncations behave, etc. Just typing "random unitary matrices" in the internet will get you a lot of results. Maybe you can ask a more specific question.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .