Parametrization of unitary matrices

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.

• Why the positive-semidefinite tag? – Dap Mar 12 at 20:13
• You obtained 2 answers. Have you read them ? – loup blanc Mar 25 at 20:12
• yes. ${}{}{{}}$ – enthdegree Mar 25 at 20:24

This link provides one method for doing this.

This link provides a different approach.

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.