On the Induced Norms of Matrices and Grothendieck problems

TL;DR

This paper develops an analytic framework to exactly compute induced matrix norms for various classes of matrices and connects these results to Grothendieck problems, advancing understanding of these complex quantities.

Contribution

It introduces a novel analytic approach to determine induced norms exactly for multiple matrix classes and links these findings to Grothendieck problems.

Findings

Exact formulas for induced norms of certain matrix classes.

Analytic solutions that bypass non-convex optimization.

Unified framework connecting matrix norms and Grothendieck problems.

Abstract

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.