connectedness of matrices with fixed singular value

Question

Let $M_n(C)$ be $n\times n$ matrix space over complex numbers. Define the set $L$ by $L=\{A\in M_n(C) : s(A)=e_1 = (1,0,\ldots ,0)\in R^n\}$, where $s(A)-$singular values of $A$ arranged in descending order. How to show that $L$ is connected subset of $M_n(C)$?

Answer 1

Let $\mathcal{U}_n$ be the space of all $n\times n$ unitary matrices. Define the function $f : \mathcal{U}_n \times \mathcal{U}_n \to M_n(C)$ by

$$ f(U,V) := U E_{11} V^* $$

Where $E_{11} := \mathrm{diag}(e_1)$. Now, $f$ is a continuous function on a connected domain. We have $L=f(\mathcal{U}_n \times\mathcal{U}_n)$. We know that the image of a connected set under a continuous function is a connected set. So $L$ is connected.