Geodesic
Combinatorial structure of $k$-semiprimitive matrix families
Sbornik. Mathematics, Tome 207 (2016) no. 5, pp. 639-651 Cet article a éte moissonné depuis la source Math-Net.Ru

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Protasov's Theorem on the combinatorial structure of $k$-primitive families of non-negative matrices is generalized to $k$-semiprimitive matrix families. The main tool is the binary relation of colour compatibility on the vertices of the coloured graph of the matrix family. Bibliography: 14 titles.
Keywords: Perron-Frobenius Theorem, coloured graphs.
Mots-clés : nonnegative matrices 
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Yu. A. Al'pin; V. S. Al'pina. Combinatorial structure of $k$-semiprimitive matrix families. Sbornik. Mathematics, Tome 207 (2016) no. 5, pp. 639-651. http://geodesic.mathdoc.fr/item/SM_2016_207_5_a0/

[1] V. Yu. Protasov, A. S. Voynov, “Sets of nonnegative matrices without positive products”, Linear Algebra Appl., 437:3 (2012), 749–765 | DOI | MR | Zbl

[2] C. Fleischhack, S. Friedland, “Asymptotic positivity of Hurwitz product traces: two proofs”, Linear Algebra Appl., 432:6 (2010), 1363–1383 | DOI | MR | Zbl

[3] E. Fornasini, M. E. Valcher, “Directed graphs, 2D state models and characteristic polynomials of irreducible matrix pairs”, Linear Algebra Appl., 263 (1997), 275–310 | DOI | MR | Zbl

[4] V. Yu. Protasov, “Classification of $k$-primitive sets of matrices”, SIAM J. Matrix Anal. Appl., 34:3 (2013), 1174–1188 | DOI | MR | Zbl

[5] D. D. Olesky, B. Shader, P. van den Driessche, “Exponents of tuples of nonnegative matrices”, Linear Algebra Appl., 356:1–3 (2002), 123–134 | DOI | MR | Zbl

[6] B. L. Shader, S. Suwilo, “Exponents of nonnegative matrix pairs”, Linear Algebra Appl., 363 (2003), 275–293 | DOI | MR | Zbl

[7] F. R. Gantmacher, The theory of matrices, v. 1, 2, Chelsea Publ., New York, 1959, x+374 pp., ix+276 pp. | MR | MR | Zbl | Zbl

[8] V. Romanovsky, “Un théorème sur les zéros des matrices non négatives”, Bull. Soc. Math. France, 61 (1933), 213–219 | MR | Zbl

[9] V. I. Romanovsky, Discrete Markov chains, Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics, Wolters–Noordhoff Publ., Groningen, 1970, xi+408 pp. | MR | Zbl | Zbl

[10] E. Fornasini, “2D Markov chains”, Linear Algebra Appl., 140 (1990), 101–127 | DOI | MR | Zbl

[11] S. N. Bernshtein, “Klassifikatsiya tsepei Markova i ikh matrits”, Sobranie sochinenii, v. IV, Nauka, M., 1964, 455–483 | Zbl

[12] Stokhasticheskaya matritsa {https://ru.wikipedia.org/wiki/Stokhasticheskaya matritsa}

[13] Yu. A. Al'pin, V. S. Al'pina, “Combinatorial properties of irreducible semigroups of nonnegative matrices”, J. Math. Sci. (N. Y.), 191:1 (2013), 4–9 | DOI | MR | Zbl

[14] A. S. Voynov, V. Yu. Protassov, “Compact noncontraction semigroups of affine operators”, Sb. Math., 206:7 (2015), 921–940 | DOI | DOI | MR | Zbl