Given a $n\times n$ unitary matrix $\mathbf{U}$, can this be rewritten as a product
$$
\mathbf{U}=\prod_{i=1}^n\mathbf{U}_i
$$
where $\mathbf{U}_i$ are unitary matrices themselves, but they only really `affect' two dimensions?
For $n=3$, they would have this shape (in analogy to the rotation matrices):
$$
\mathbf{U}_1=
\pmatrix{
a_{11}&a_{12}&0\\
a_{21}&a_{22}&0\\
0&0&1},
\quad
\mathbf{U}_2=
\pmatrix{
b_{11}&0&b_{12}\\
0&1&0\\
b_{21}&0&b_{22}},
\mathbf{U}_3=
\pmatrix{
1&0&0\\
0&c_{11}&c_{12}\\
0&c_{21}&c_{22}}
$$
With the 2×2 unitary matrices $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{C}$. (Maybe, the ones have to be replaced with a phase factor $e^{i\varphi}$ to have enough flexibility?)
The point is, that I have a property which is preserved under multiplication with such a "2D" unitary matrix. And I suspect that it generalizes to any unitary matrix (ideally even a unitary transformation in $L^2$), but I am not sure.
Best Answer
Consider the map $(U_2)^{n(n-1)/2} \to U_n$ sending a set of "2D" unitary matrices for all choices of the 2 dimensions to the product unitary matrix. It's derivative at $(Id)^{n(n-1)/2}$ is surjective (it sends a 2D skew-Hermitian matrix in the tangent space of the $k$th $U_2$ to the same 2D skew-Hermitian matrix in the selected dimensions, and any skew-Hermitian matrix is a sum of those). Thus the map itself is locally onto (implicit function theorem), so any unitary matrix sufficiently close to $Id$ in $U_n$ is a product of the type you want. But then any unitary matrix is (a neighbourhood of identity $U$ generates $G$ where $G$ is a connected lie group).