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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Ya. B. Lopatinskii, “Factorization of a polynomial matrix,” Nauch. Zap. L’vov. Politekhn. Inst., 38, No. 2, 3–7 (1956).
P. S. Kazimirs’kyi, Factorization of Matrix Polynomials [in Ukrainian], Naukova Dumka, Kiev (1981).
I. Gohberg, P. Lankaster, and L. Rodman, Matrix Polynomials, Academic Press, New York (1982).
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).
P. S. Kazimirskii and M. N. Urbanovich, “On factorization of a matrix binomial,” Ukr. Mat. Zh., 24, No. 4, 454–464 (1973).
A. S. Markus and I. V. Mereutsa, “On some properties of simple λ-matrices,” Mat. Issled., 10, No. 3, 207–213 (1975).
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.
L. A. Sakhnovich, “On factorization of transfer operator functions,” Dokl. Akad. Nauk SSSR, 226, No. 4, 781–784 (1976).
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).
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.
B. Z. Shavarovskii, Similarity Transformations of Matrix Polynomials and Their Factorization [in Russian], Candidate-Degree Thesis (Physics and Mathematics), L’vov (1985).
I. N. Krupnik, “On decomposition of a matrix pencil into a product of linear factors,” Mat. Zametki, 49, No. 2, 95–101 (1991).
I. Krupnik, “Decomposition of a monic polynomial into a product of linear factors,” Lin. Alg. Appl., 167, 239–242 (1992).
B. Z. Shavarovskii, “On factorizable polynomial matrices,” Mat. Zametki, 68, No. 4, 593–607 (2000).
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).
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).
F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).
M. Newman, “On the Smith normal form,” J. Res. Bur. Stand. Sect., 75, 81–84 (1971).
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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