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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.03772 |
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| _version_ | 1866914531234545664 |
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| author | Truong, Lan V. Duong, M. H. |
| author_facet | Truong, Lan V. Duong, M. H. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03772 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Induced Norms of Matrices and Grothendieck problems Truong, Lan V. Duong, M. H. Optimization and Control Computational Complexity 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. |
| title | On the Induced Norms of Matrices and Grothendieck problems |
| topic | Optimization and Control Computational Complexity |
| url | https://arxiv.org/abs/2605.03772 |