Geodesic
On common unital divisors of matrix polynomials over an arbitrary field
Sbornik. Mathematics, Tome 78 (1994) no. 2, pp. 427-435 
V. M. Prokip. On common unital divisors of matrix polynomials over an arbitrary field. Sbornik. Mathematics, Tome 78 (1994) no. 2, pp. 427-435. http://geodesic.mathdoc.fr/item/SM_1994_78_2_a9/
@article{SM_1994_78_2_a9,
     author = {V. M. Prokip},
     title = {On common unital divisors of matrix polynomials over an~arbitrary field},
     journal = {Sbornik. Mathematics},
     pages = {427--435},
     year = {1994},
     volume = {78},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1994_78_2_a9/}
}
TY  - JOUR
AU  - V. M. Prokip
TI  - On common unital divisors of matrix polynomials over an arbitrary field
JO  - Sbornik. Mathematics
PY  - 1994
SP  - 427
EP  - 435
VL  - 78
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1994_78_2_a9/
LA  - en
ID  - SM_1994_78_2_a9
ER  - 
%0 Journal Article
%A V. M. Prokip
%T On common unital divisors of matrix polynomials over an arbitrary field
%J Sbornik. Mathematics
%D 1994
%P 427-435
%V 78
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1994_78_2_a9/
%G en
%F SM_1994_78_2_a9

Voir la notice de l'article provenant de la source Math-Net.Ru

The problem of common left divisors of matrix polynomials over an arbitrary field is investigated. A necessary and, with certain restrictions, sufficient condition is established for the existence of a common unital divisor with a prescribed characteristic polynomial of two nonsingular matrix polynomials. The approach offered in this paper has made it possible to solve this problem completely for matrix polynomials if at least one of them has pairwise relatively prime elementary divisors. In the case when the desired divisor exists, a method of constructing it is specified.

[1] Kurosh A. G., Kurs vysshei algebry, Nauka, M., 1971

[2] Barnett S., “Regular polynomial matrices having relatively prime determinants”, Proc. Cambridge Philos. Soc., 65 (1969), 585–590 | DOI | MR | Zbl

[3] Barnett S., “Regular greatest common divisors of two polynomial matrices”, Proc. Cambridge Philos. Soc., 72 (1972), 161–165 | DOI | MR | Zbl

[4] Gokhberg I. Ts., Khainig G., “Rezultantnaya matritsa i ee obobscheniya. I: Rezultantnyi operator matrichnykh polinomov”, Acta Sci. Math., 37:1/2 (1975), 41–61 | Zbl

[5] Petrichkovich V. M., Prokip V. M., “Ob obschikh delitelyakh matrichnykh mnogochlenov”, Matem. metody i fiz.-mekh. polya, 1983, no. 18, 23–26 | MR | Zbl

[6] Gohberg J., Lancaster P., Rodman L., Matrix Polynomials, Acad. Press, New York–London, 1982 | MR

[7] Lancaster P., “Common eigenvalue, divisors and multiples of matrix polynomials”, Linear Algebra and its Appl., 84 (1986), 139–160 | DOI | MR | Zbl

[8] Petrichkovich V. M., Prokip V. M., “O faktorizatsii mnogochlennykh matrits nad proizvolnym polem”, Ukr. matem. zhurn., 38:4 (1986), 478–483 | MR | Zbl

[9] Prokip V. M., “O edinstvennosti unitalnogo delitelya matrichnogo mnogochlena”, Tezisy soobsch. VI-go simpoz. po teorii kolets, algebr i modulei (Lvov, 11–13 sentyabrya 1990 g.), 106