# What is the eigenvalue distribution of arbitrary unitary matrices?

I had a question regarding the nature of the eigenvalue distribution of unitary matrices.

Searching for the answer I found that the unitary matrices which are sampled randomly have a defined eigenvalue distribution. What I would like to know is that can we comment on the distribution of the eigenvalues for any arbitrary unitary matrix too? It would be really great if someone could point to some resources for this. Thanks!

If you have a fixed unitary matrix $$U$$ then you can have absolutely any distribution of eigenvalues that lie on the unit circle in $$\mathbb{C}$$. For instance $$\alpha I$$ for any complex number $$\alpha$$ with $$|\alpha|=1$$ is a unitary matrix. It has $$n$$ eigenvalues all equal to $$\alpha$$.
If you don't like degeneracy then take any $$n$$ distinct $$\alpha_i$$ on the unit circle and then $$\mathrm{Diag}(\alpha_1, \dots, \alpha_n)$$, the diagonal matrix with $$\alpha_1,\dots,\alpha_n$$ on the diagonal is a unitary matrix with eigenvalues $$\alpha_1, \dots, \alpha_n$$.
If you don't like diagonal matrices then take your favorite unitary $$V$$ and then $$V \mathrm{Diag}(\alpha_1, \dots, \alpha_n) V^*$$ is another unitary matrix with eigenvalues $$\alpha_1,\dots, \alpha_n$$.