Finding a complete set of solutions or proving unsolvability for certain classes of matrix polynomial equations with commuting coefficients
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 12, pp. 1988-1997
B. Z. Shavarovskii. Finding a complete set of solutions or proving unsolvability for certain classes of matrix polynomial equations with commuting coefficients. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 47 (2007) no. 12, pp. 1988-1997. http://geodesic.mathdoc.fr/item/ZVMMF_2007_47_12_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

Two classes of matrix polynomial equations with commuting coefficients are examined. It is shown that the equations in one class have complete sets of solutions, whereas the equations in the other class are unsolvable. A method is given for finding the solution set of an equation in the former class.

[1] Kozlov V. V., “Lineinye sistemy s kvadratichnym integralom i simplekticheskaya geometriya prostranstv Artina”, Prikl. matem. i mekhan., 68:3 (2004), 371–383 | MR | Zbl

[2] Kozlov V. V., “Ogranicheniya kvadratichnykh form na lagranzhevy ploskosti, kvadratnye matrichnye uravneniya i giroskopicheskaya stabilizatsiya”, Funkts. analiz i ego prilozh., 39:4 (2005), 32–47 | MR | Zbl

[3] Ikramov Kh. D., Chislennoe reshenie matrichnykh uravnenii, Nauka, M., 1984 | MR | Zbl

[4] Lancaster P., Rodman L., Algebraic Riccati equations, Clarendon Press, Oxford, 1995 | MR | Zbl

[5] Aliev F. A., Larin V. B., “Osobye sluchai v zadachakh optimizatsii statsionarnykh lineinykh sistem, funktsioniruyuschikh po printsipu obratnoi svyazi”, Prikl. mekhan., 39:3 (2003), 3–25 | MR | Zbl

[6] Gelfand S. I., “O chisle reshenii kvadratnogo uravneniya”, Globus. Obschematem. seminar, v. 1, MTsNMO, M., 2004, 124–133

[7] Kozlov V V., Obschaya teoriya vikhrei, Izd. dom “Udmurtskii un-t”, Izhevsk, 1998 | MR | Zbl

[8] Kirillov A. A., “O chisle reshenii uravneniya $X^2=0$ v treugolnykh matritsakh nad konechnym polem”, Funkts. analiz i ego prilozh., 29:1 (1995), 82–87 | MR | Zbl

[9] Markus A. C., Mereutsa I. V., “O polnom nabore kornei operatornogo uravneniya, sootvetstvuyuschego polinomialnomu operatornomu puchku”, Izv. AN SSSR. Ser. matem., 37:5 (1973), 1108–1131 | MR | Zbl

[10] Dennis J. E., Traub J. P., Weber R. P., “The algebraic theory of matrix polynomials”, SIAM J. Numer. Analys., 13:6 (1976), 831–845 | DOI | MR | Zbl

[11] Gantmakher F. R., Teoriya matrits, Nauka, M., 1967 | MR

[12] Newman M., “On the Smith normal form”, J. Res. Bur. Standards Sect., 75 (1971), 81–84 | MR | Zbl

[13] Kazimipckii P. S., Rozklad matrichnikh mnogochleniv na mnozhniki, Nauk. dumka, Kiïv, 1981