We study the induced matrix norm $\|\bA\|_{q \to r}$, whose exact value has been known only in a few classical cases. Determining this norm has long been regarded as difficult due to the highly non-convex nature of its variational definition. Existing works offer numerical estimates or analytic bounds but no exact formula. In this paper we present a purely analytic framework that determines $\|\bA\|_{q \to r}$ exactly for all $q, r \ge 1$ for several classes of important matrices. For these matrices, using a direct connection between the induced norms and Grothendieck problems, our results also simultaneously provide exact values for the later.

翻译：我们研究诱导矩阵范数$\|\bA\|_{q \to r}$，其精确值仅在少数经典情形下已知。由于该范数的变分定义具有高度非凸性，确定其值长期被视为困难问题。现有工作仅提供数值估计或解析界限，缺乏精确公式。本文提出一个纯解析框架，可精确确定若干重要矩阵类在所有$q, r \ge 1$情形下的$\|\bA\|_{q \to r}$值。对于这些矩阵，利用诱导范数与Grothendieck问题之间的直接联系，我们的结果同时也为后者提供了精确值。