Let $n\in\mathbb N$. Let $I_n:\mathbb C^n\to\mathbb C^n$ denote the identity [linear map](/page/Linear%20Map), and let $U(n)$ denote the group of unitary linear maps on $\mathbb C^n$ with respect to the standard Hermitian [inner product](/page/Inner%20Product). Let $A\le U(n)$ be an abelian subgroup. Define
paragraph
admin
\begin{align*}
T_{U(n)}:=\{\operatorname{diag}(z_1,\dots,z_n)\in U(n): z_i\in U(1)\text{ for }1\le i\le n\}
\end{align*}
latex_env
admin
to be the diagonal torus of $U(n)$. Then there exists $u\in U(n)$ such that