I'm looking for an approximation of a given matrix by an injective matrix close enough to the given matrix.
Suppose that $U \subset \mathbb{R}^{n}$ is convex compact and a matrix $A \in \mathbb{R}^{m \times n}$ with $m > n$ is given. For any $\epsilon > 0$, can one always find an injective matrix $B$ such that $\lVert (A-B)u \rVert < \epsilon$ for any $u \in U$ and $u_{1} \neq u_{2}$ $\Rightarrow$ $Bu_{1} \neq Bu_{2}$?
- Note that it is sufficient to show that such $B$ exists with $\lVert A - B \rVert < \epsilon$ (induced norm) as $\lVert(A-B) u\rVert \leq \lVert A-B \rVert \lVert u \rVert \leq \lVert A-B \rVert C$ for some $C >0$.