On the Spectral Region of 4-Cycle Stochastic Matrices
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911658774888448 |
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| author | Vagenende, Brando Verbeken, Brecht Algaba, Andres Guerry, Marie-Anne |
| author_facet | Vagenende, Brando Verbeken, Brecht Algaba, Andres Guerry, Marie-Anne |
| contents | We study the spectrum of 4-cycle row-stochastic matrices. For real eigenvalues the spectral region is [-1,1]. For nonreal eigenvalues a+ib we derive necessary conditions in terms of the real and imaginary parts, including the inequality a+|b| <= 1 and the condition (b^2+a^2+a)^2+2a^2-b^2 >= 0. We also prove conversely that every point in the corresponding interior region occurs as an eigenvalue of a 4-cycle matrix. The proof is organized through a reformulation of the characteristic equation, an argument parametrization, a convex-analytic criterion, and explicit boundary constructions. Hence, the spectral region for the 4-cycle row-stochastic matrices is exactly and explicitly determined. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06743 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Spectral Region of 4-Cycle Stochastic Matrices Vagenende, Brando Verbeken, Brecht Algaba, Andres Guerry, Marie-Anne Spectral Theory Probability 15A18, 15B51 We study the spectrum of 4-cycle row-stochastic matrices. For real eigenvalues the spectral region is [-1,1]. For nonreal eigenvalues a+ib we derive necessary conditions in terms of the real and imaginary parts, including the inequality a+|b| <= 1 and the condition (b^2+a^2+a)^2+2a^2-b^2 >= 0. We also prove conversely that every point in the corresponding interior region occurs as an eigenvalue of a 4-cycle matrix. The proof is organized through a reformulation of the characteristic equation, an argument parametrization, a convex-analytic criterion, and explicit boundary constructions. Hence, the spectral region for the 4-cycle row-stochastic matrices is exactly and explicitly determined. |
| title | On the Spectral Region of 4-Cycle Stochastic Matrices |
| topic | Spectral Theory Probability 15A18, 15B51 |
| url | https://arxiv.org/abs/2605.06743 |