The nonsingular matrix closest to singular one

Question

It is well known that given nonsingular matrix $A$, the distance to closest singular is given by $\|A^{-1}\|^{-1}$.

My question is given singular matrix $B$ what is the distance to closest nonsingular? Is it the same as first problem?

Answer 1

There is nonsingular matrix at distance at most $\epsilon$ for every $\epsilon >0$.

Indeed, let $B$ be any matrix, then $p(t)=\det(B-t I)$ is polynomial of $t$ and has finitely many roots. In particular, there exists $\tau>0$ such that $p(t)\neq 0$ for all $t\in (0,\tau)$ and thus $B_t = B-t I$ is nonsingular for all $t\in(0,\tau)$. Now, for every $\epsilon >0$, you can choose $t\in (0,\tau)$ small enough so that $\|B-B_t\|=t\|I\|< \epsilon$.