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arXiv · 2605.03772

On the Induced Norms of Matrices and Grothendieck problems

Lan V. Truong·M. H. Duong
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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.

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Lan V. Truong, M. H. Duong. 2026-05-05. On the Induced Norms of Matrices and Grothendieck problems. https://arxiv.org/abs/2605.03772

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