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The eigenvalues of a unitary matrix lie on the unit circle. What are some applications in which the eigenvalue distribution of the matrix is important? For instance, that the eigenvalues are clustered, that they live in some half plane, etc. I'm particularly interested in distributional properties which is "global" in the sense moving a few eigenvalues (potentially arbitrarily far) or many eigenvalues a small distance doesn't really change the meaning. So for instance, that the matrix is in $SO(n)$ is not such a property because we could move a single eigenvalue from $-1$ to $1$ to change this property.

Likewise, an explication that the eigenvalues do not have meaning is also useful. For instance, I know that the unitary matrices from many factorization are only unique up to scaling by diagonal matrices with entries on the unit circle. Do such scaling preserve any meaningful properties of the spectrum?

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  • $\begingroup$ Are you asking specifically about unitary matrices, or any type of matrix? $\endgroup$
    – JimmyK4542
    Commented Jun 15, 2021 at 4:46
  • $\begingroup$ Yes, just unitary $\endgroup$
    – overfull hbox
    Commented Jun 15, 2021 at 4:47
  • $\begingroup$ Oh ok. I had a few examples from signal processing/control theory where it is important for a matrix to have all of it's eigenvalues inside the unit circle, or all of it's eigenvalues in the left-half of the complex plane. But of course, these aren't unitary. $\endgroup$
    – JimmyK4542
    Commented Jun 15, 2021 at 4:49

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This is just a discussion and if you are satisfied, only then consider this as an answer.

Eigen values of a matrix contains various information about the matrix we all know it. But what we don't know is that what actually matrices do to our universe. This topic is so big that I can not explain by covering all the possible use of unitary matrices but I can show you a path on which you can travel and explore by yourself.

This is just a quote copied from Wikipedia : "In the Standard Model of particle physics, the Cabibbo–Kobayashi–Maskawa matrix, CKM matrix, quark mixing matrix, or KM matrix is a unitary matrix which contains information on the strength of the flavour-changing weak interaction. Technically, it specifies the mismatch of quantum states of quarks when they propagate freely and when they take part in the weak interactions. It is important in the understanding of CP violation. This matrix was introduced for three generations of quarks by Makoto Kobayashi and Toshihide Maskawa, adding one generation to the matrix previously introduced by Nicola Cabibbo. This matrix is also an extension of the GIM mechanism, which only includes two of the three current families of quarks."

I would highly recommend you to go there and check all the references. They contain some rich topics on unitary matrices.

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