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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!

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No this is not possible.

When people talk about the distribution of eigenvalues they mean the expected eigenvalues when the unitary matrices are sampled with respect to some measure.

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 $\alpha_1,\dots, \alpha_n$.

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  • $\begingroup$ What I was actually looking for was a way to somehow 'constrain' my initial search space for a minimum eigenvalue solver. Thank you for pointing out the flaw in this approach though ! $\endgroup$ Jun 23 at 14:32

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