We consider the following extremal problem: Among all matrices that map a given point x of the Euclidean space \(\mathbb {R}^n\) to a given point b of the Euclidean space \(\mathbb {R}^m\) , find a matrix of minimal norm. The article uses the Hölder norm of vectors and matrices depending on a parameter p. It is shown that for every \(p \in (1, +\infty )\) there exists a unique solution of the stated problem and an explicit formula for it is derived. Limits of the optimal matrix are found as p approaches the boundary values \(p \rightarrow 1\) and \(p \rightarrow +\infty \) . It is established that the limiting matrices are solutions of the corresponding limiting extremal problems. However, unlike the case \(p \in (1, +\infty )\) , uniqueness of these limiting solutions is not guaranteed. A full description of the entire set of solutions of the nonsmooth limiting problems is given.
