On the Induced Norms of Matrices and Grothendieck problems
For researchers in functional analysis and optimization, this provides exact formulas for previously intractable norms, though limited to specific matrix classes.
The paper presents an analytic framework that exactly determines the induced matrix norm for several important matrix classes across all q, r ≥ 1, also yielding exact values for related Grothendieck problems.
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.