Publication: Normal Tropical (0,−1)-Matrices and Their Orthogonal Sets
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2023-02
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Abstract
Square matrices A and B are orthogonal if AʘB = Z = BʘA, where Z is the matrix entries equal to 0, and ʘ is the tropical matrix multiplication. We study orthogonality for normal matrices over the set {0, −1}, endowed with tropical addition and multiplication. To do this, we investigate the orthogonal set of a matrix A, i.e., the set of all matrices orthogonal to A. In particular, we study the family of minimal elements inside the orthogonal set, called a basis. Orthogonal sets and bases are computed for various matrices and matrix sets. Matrices whose bases are singletons are characterized. Orthogonality and minimal orthogonality are described in the language of graphs. The geometric interpretation of the results obtained is discussed.
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 24, No. 1, pp. 5–30, 2022.
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1. B. Bakhadly, A. Guterman, and M. J. de la Puente, “Orthogonality for (0, −1) tropical normal matrices,” Special Matrices, 8, 40–60 (2020).
2. G. Birkhoff, Lattice Theory, Providence: Rhode Island (1967).
3. P. Butkovic, “Simple image set of (max, +) linear mappings,” Discrete Appl. Math., 105, 73–86 (2000).
4. P. Butkovic, Max-plus Linear Systems: Theory and Algorithms, Springer (2010).
5. P. Crawley and R. P. Dilworth, Algebraic Theory of Lattices, Prentice Hall (1973).
6. M. Develin and B. Sturmfels, “Tropical convexity,” Doc. Math. 9, 1–27 (2004), Erratum in Doc. Math., 9, 205–206 (2004).
7. I. B. Kozhukhov and A. V. Mikhalev, “Polygons over semigroups: selected questions of structure theory,” Fundam. Prikl. Mat., textbf23, No. 3, 141–199 (2020).
8. G. L. Litvinov and S. N. Sergeev, eds., Tropical and Idempotent Mathematics, Contemp. Math., Vol. 495, Amer. Math. Soc. (2009).
9. M. J. de la Puente, “Tropical linear maps on the plane,” Linear Algebra Appl., 435, No. 7, 1681–1710 (2011).
10. M. J. de la Puente, “On tropical Kleene star matrices and alcoved polytopes,” Kybernetika, 49, No. 6, 897–910 (2013).
11. M. J. de la Puente, “Quasi-Euclidean classification of alcoved convex polyhedra,” Linear Multilinear Algebra, 67 (2019).
12. S. Sergeev, “Multiorder, Kleene stars and cyclic projectors in the geometry of max cones,” in: G. L. Litvinov and S. N. Sergeev, eds., Tropical and Idempotent Mathematics, Contemp. Math., Vol. 495, Amer. Math. Soc (2009), pp. 317–342.
13. S. Sergeev, H. Scheneider, and P. Butkovic, “On visualization, subeigenvectors and Kleene stars in max algebra,” Linear Algebra Appl., 431, 2395–2406 (2009).
14. N. M. Tran, “Enumerating polytropes,” J. Combin. Theory Ser. A, 151, 1–22 (2017).
15. M. Yoeli, “A note on a generalization of Boolean matrix theory,” Am. Math. Mon., 68, No. 6, 552–557 (1961).