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Does anyone know a simple way to parametrize the space of $n\times n$ complex unitary matrices into a set of independent complex numbers in some complex-rectangle? Specifically the mapping and inverse mapping that does this.

Like a generalized euler angle.

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  • $\begingroup$ Why the positive-semidefinite tag? $\endgroup$ – Dap Mar 12 at 20:13
  • $\begingroup$ You obtained 2 answers. Have you read them ? $\endgroup$ – loup blanc Mar 25 at 20:12
  • $\begingroup$ yes. ${}{}{{}}$ $\endgroup$ – enthdegree Mar 25 at 20:24
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This link provides one method for doing this.

This link provides a different approach.

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The Cayley transform is valid for unitary matrices.

https://en.wikipedia.org/wiki/Cayley_transform#Matrix_map

$f:A\in SK_n\mapsto (I-A)(I+A)^{-1}\in U_{-1}(n)$ is a diffeomorphism from the set of skew-hermitian matrices to the set of unitary matrices that does not admit $-1$ as eigenvalue. Clearly, $f^{-1}:U\mapsto (I-U)(I+U)^{-1}$.

Note that $SK_n$ is a real vector space of dimension $n^2$. Then one has a (partial) parametrization of $U(n)$ with $n^2$ parameters, the dimension of the algebraic set $U(n)$ over $\mathbb{R}$.

Remark that $U(n)$ cannot be globally parametrized by $SK_n$ because $\pi_1(U(n))=\pi_1(S^1)=\mathbb{Z}$. You have to make a hole in $U(n)$ to be able to flatten it.


EDIT: Conclusion. There is no algebraic parametrization of $U(n)$ over $\mathbb{C}^{n^2}$ because $U(n)$ is not an affine algebraic variety in complex space $\mathbb{C}^{n^2}$. We can see that, noting that $U(n)$ is compact OR noting that $U(n)$ cannot be defined by a system of polynomial equations in the entries of the matrix.

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