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Decomposability of matrix polynomials with commuting coefficients into a product of linear factors

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Ukrainian Mathematical Journal Aims and scope

The list of known sets of factorizable matrix polynomials is supplemented by new sets of polynomials of this sort. The known set of nonfactorizable matrix polynomials is extended. These results can be applied to the study of polynomial equations and systems of differential equations with constant coefficients.

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References

  1. Ya. B. Lopatinskii, “Factorization of a polynomial matrix,” Nauch. Zap. L’vov. Politekhn. Inst., 38, No. 2, 3–7 (1956).

    MathSciNet  Google Scholar 

  2. P. S. Kazimirs’kyi, Factorization of Matrix Polynomials [in Ukrainian], Naukova Dumka, Kiev (1981).

    Google Scholar 

  3. I. Gohberg, P. Lankaster, and L. Rodman, Matrix Polynomials, Academic Press, New York (1982).

    MATH  Google Scholar 

  4. P. S. Kazimirskii, “Solution of the problem of separation of a regular factor from a matrix polynomial,” Ukr. Mat. Zh., 32, No. 4, 483–498 (1980).

    MathSciNet  Google Scholar 

  5. P. S. Kazimirskii and M. N. Urbanovich, “On factorization of a matrix binomial,” Ukr. Mat. Zh., 24, No. 4, 454–464 (1973).

    Google Scholar 

  6. A. S. Markus and I. V. Mereutsa, “On some properties of simple λ-matrices,” Mat. Issled., 10, No. 3, 207–213 (1975).

    MATH  MathSciNet  Google Scholar 

  7. P. S. Kazimirskii, “Separation of a regular linear factor of simple structure from a matrix polynomial,” in: Theoretical and Applied Problems of Algebra and Differential Equations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1976), pp. 29–40.

  8. L. A. Sakhnovich, “On factorization of transfer operator functions,” Dokl. Akad. Nauk SSSR, 226, No. 4, 781–784 (1976).

    MathSciNet  Google Scholar 

  9. P. S. Kazimirskii and V. M. Petrichkovich, “Decomposability of polynomial matrices into a product of linear factors,” Mat. Met. Fiz.-Mekh. Polya, 8, 3–9 (1978).

    Google Scholar 

  10. P. S. Kazimirskii and V. M. Petrichkovich, “One sufficient condition for the decomposability of a matrix square trinomial into a product of linear factors,” in: Proceedings of the Second All-Union Symposium on the Theory of Rings, Algebras, and Moduli [in Russian], Shtiintsa, Kishenev (1974), pp. 29–30.

  11. B. Z. Shavarovskii, Similarity Transformations of Matrix Polynomials and Their Factorization [in Russian], Candidate-Degree Thesis (Physics and Mathematics), L’vov (1985).

  12. I. N. Krupnik, “On decomposition of a matrix pencil into a product of linear factors,” Mat. Zametki, 49, No. 2, 95–101 (1991).

    MathSciNet  Google Scholar 

  13. I. Krupnik, “Decomposition of a monic polynomial into a product of linear factors,” Lin. Alg. Appl., 167, 239–242 (1992).

    Article  MathSciNet  Google Scholar 

  14. B. Z. Shavarovskii, “On factorizable polynomial matrices,” Mat. Zametki, 68, No. 4, 593–607 (2000).

    MathSciNet  Google Scholar 

  15. V. M. Petrychkovych, “On multiplicity of characteristic roots, degrees of elementary divisors, and factorization of polynomial matrices,” Mat. Met. Fiz.-Mekh. Polya, 48, No. 2, 7–17 (2005).

    MATH  MathSciNet  Google Scholar 

  16. V. R. Zelisko and B. Z. Shavarovskii, “Decomposition of a matrix polynomial into a product of factors of simple structure,” Mat. Met. Fiz.-Mekh. Polya, 15, 43–48 (1982).

    MATH  MathSciNet  Google Scholar 

  17. F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).

    MATH  Google Scholar 

  18. M. Newman, “On the Smith normal form,” J. Res. Bur. Stand. Sect., 75, 81–84 (1971).

    MATH  Google Scholar 

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 8, pp. 1114–1123, August, 2010.

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Shavarovs’kyi, B.Z. Decomposability of matrix polynomials with commuting coefficients into a product of linear factors. Ukr Math J 62, 1295–1306 (2011). https://doi.org/10.1007/s11253-011-0430-2

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  • DOI: https://doi.org/10.1007/s11253-011-0430-2

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