Geodesic
Similarity transformations of decomposable matrix polynomials and related questions
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 9, pp. 1539-1553 Cet article a éte moissonné depuis la source Math-Net.Ru

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A elationship is found between the similarity transformations of decomposable matrix polynomials with relatively prime elementary divisors and the equivalence transformations of the corresponding matrices with scalar entries. Matrices with scalar entries are classified with respect to equivalence transformations based on direct sums of lower triangular almost Toeplitz matrices. This solves the similarity problem for a special class of finite matrix sets over the field of complex numbers. Eventually, this problem reduces to the one of special diagonal equivalence between matrices. Invariants of this equivalence are found. 
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B. Z. Shavarovskii. Similarity transformations of decomposable matrix polynomials and related questions. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 49 (2009) no. 9, pp. 1539-1553. http://geodesic.mathdoc.fr/item/ZVMMF_2009_49_9_a0/

[1] Gelfand I. M., Ponomarev V. A., “Nerazlozhimye predstavleniya gruppy Lorentsa”, Uspekhi matem. nauk, 23:2 (1968), 3–59 | MR

[2] Drozd Yu. A., “Derived tame and derived wild algebras”, Algebra and Discrete Math., 1 (2004), 57–74 | MR | Zbl

[3] Bondarenko V. M., “Linear operators on $S$-graded vector spaces”, Linear Algebra and Appl., 365 (2003), 45–90 | DOI | MR | Zbl

[4] Bondarenko V. M., “Pro klasifkatsiyu medulyarnikh uzagalnenikh grup kvaternioniv”, Dop. Nats. akademiï nauk Ukraïni, 11 (2004), 11–16 | MR | Zbl

[5] Drozd Yu. A., “O ruchnykh i dikikh matrichnykh zadachakh”, Matrichnye zadachi, IM AN USSR, Kiev, 1977, 104–114 | MR

[6] Friedland S., “Simultaneous similarity of matrices”, Advances. Math., 50 (1983), 189–265 | DOI | MR | Zbl

[7] Belitskii G. R., “Normal forms in matrix spaces”, Integral Equations Operator Theory, 30:3 (2000), 251–283 | DOI | MR

[8] Sergeichuk V. V., Canonical matrices and related questions, Inst. Math. NAS Ukraine, Kyiv, 2006 | Zbl

[9] Shavarovskii B. Z., “Reduktsiya matrits pri pomoschi ekvivalentnykh i podobnykh preobrazovanii”, Matem. zametki, 64:5 (1998), 769–782 | MR

[10] Shavarovskii B. Z., “Polnaya sistema invariantov pary matrits chetnogo poryadka so spetsialnoi formoi Smita ee kharakteristicheskoi matritsy”, Matem. zametki, 73:6 (2003), 923–941 | MR

[11] Shavarovskii B. Z., “O nekotorykh “ruchnykh” i “dikikh” aspektakh problemy poluskalyarnoi ekvivalentnosti mnogochlennykh matrits”, Matem. zametki, 76:1 (2004), 119–132 | MR

[12] Shavarovskii B. Z., “O podobii par matrits chetnogo poryadka”, Matem. zametki, 81:3 (2007), 448–463 | MR | Zbl

[13] Dias da Silva J. A., Laffey T. J., “On simultaneous similarity of matrices and related questions”, Linear Algebra and its Applic., 291 (1999), 167–184 | DOI | MR | Zbl

[14] Shavarovskii B. Z., “Razreshimost matrichnykh uravnenii v koltsakh kvazidiagonalnykh matrits i podobie matrichnykh mnogochlenov”, Zh. vychisl. matem. i matem. fiz., 46:8 (2006), 1353–1362 | MR

[15] Kazimirskii P. S., Rozklad matrichnikh mnogochleniv na mnozhniki, Naukova dumka, Kiïv, 1981

[16] Gochberg I., Lancaster P., Rodman L., Matrix polynomials, Acad. Press, New York, 1982 | MR

[17] Markus A. C., Mereutsa I. V., “O nekotorykh svoistvakh prostykh $\lambda$-matrits”, Matem. issled., 10:3 (1975), 207–213 | MR | Zbl

[18] Sakhnovin L. A., “O faktorizatsii peredatochnykh operator-funktsii”, Dokl. AN SSSR, 226:4 (1976), 781–784 | MR

[19] Ikramov Kh. D., Zadachnik po lineinoi algebre, Nauka, M., 1975 | MR

[20] Shavarovskii B. Z., “Poluskalyarnaya ekvivalentnost polinomialnykh matrits s poparno razlichnymi kornyami ikh kharakteristicheskogo mnogochlena”, Matem. metody i fiz.-mekhan. polya, 19 (1984), 33–37 | MR