How to prove **using SVD** that $\mathbb{C}^{n \times n}$ is dense for nonsingular matrices? How to show using SVD that the set of nonsingular matrices is dense in $\mathbb{C}^{n \times n}$? That is, for any $A \in \mathbb{C}^{n \times n}$, and any given $\varepsilon  > 0$, there exists a nonsingular matrix $A_\varepsilon \in \mathbb{C}^{n \times n}$ such that:
$\left \| A-A_\varepsilon \right \| \le \varepsilon$.

Let $A=U\Sigma V^\ast$ be a full SVD of $A$. Just modify the zero diagonal entries of $\Sigma$ to small nonzero ones.
