Geodesic
On Some “Tame” and “Wild” Aspects of the Problem of Semiscalar Equivalence of Polynomial Matrices
Matematičeskie zametki, Tome 76 (2004) no. 1, pp. 119-132 
B. Z. Shavarovskii. On Some “Tame” and “Wild” Aspects of the Problem of Semiscalar Equivalence of Polynomial Matrices. Matematičeskie zametki, Tome 76 (2004) no. 1, pp. 119-132. http://geodesic.mathdoc.fr/item/MZM_2004_76_1_a11/
@article{MZM_2004_76_1_a11,
     author = {B. Z. Shavarovskii},
     title = {On {Some} {{\textquotedblleft}Tame{\textquotedblright}} and {{\textquotedblleft}Wild{\textquotedblright}} {Aspects} of the {Problem} of {Semiscalar} {Equivalence} of {Polynomial} {Matrices}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {119--132},
     year = {2004},
     volume = {76},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_1_a11/}
}
TY  - JOUR
AU  - B. Z. Shavarovskii
TI  - On Some “Tame” and “Wild” Aspects of the Problem of Semiscalar Equivalence of Polynomial Matrices
JO  - Matematičeskie zametki
PY  - 2004
SP  - 119
EP  - 132
VL  - 76
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MZM_2004_76_1_a11/
LA  - ru
ID  - MZM_2004_76_1_a11
ER  - 
%0 Journal Article
%A B. Z. Shavarovskii
%T On Some “Tame” and “Wild” Aspects of the Problem of Semiscalar Equivalence of Polynomial Matrices
%J Matematičeskie zametki
%D 2004
%P 119-132
%V 76
%N 1
%U http://geodesic.mathdoc.fr/item/MZM_2004_76_1_a11/
%G ru
%F MZM_2004_76_1_a11

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

The problem of reducing polynomial matrices to canonical form by using semiscalar equivalent transformations is studied. This problem is wild as a whole. However, it is tame in some special cases. In the paper, classes of polynomial matrices are singled out for which canonical forms with respect to semiscalar equivalence are indicated. We use this tool to construct a canonical form for the families of coefficients corresponding to the polynomial matrices. This form enables one to solve the classification problem for families of numerical matrices up to similarity.

[1] Kazimirskii P. S., Petrichkovich V. M., “Pro ekvivalentnist polinomialnikh matrits”, Teoretichni ta prikladni pitannya algebri i diferentsialnikh rivnyan, Naukova dumka, Kiïv, 1977, 61–66

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

[3] Donovan P., Freislich M. R., “Some evidence for an extension of the Brauer–Thrall conjecture”, Sonderforschungsbereich Theor. Math., 40 (1973), 24–26

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

[5] Gelfand I. M., Ponomarev V. A., “Nerazlozhimye predstavleniya gruppy Lorentsa”, UMN, 23:2 (1968), 3–60 | MR | Zbl

[6] Nazarova L. A., Roiter A. V., “Ob odnoi zadache I. M. Gelfanda”, Funktsion. analiz i ego prilozh., 7:4 (1973), 54–69 | MR | Zbl

[7] Bondarenko V. M., “Predstavleniya diedralnykh grupp nad polem kharakteristiki 2”, Matem. sb., 96:1 (1975), 63–74 | MR | Zbl

[8] Bondarenko V. M., Drozd Yu. A., “Predstavlencheskii tip konechnykh grupp”, Zapiski nauch. sem. LOMI, 71, Nauka, L., 1977, 24–41 | MR | Zbl

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

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

[11] Sergeichuk V. V., “Canonical matrices for linear matrix problems”, Linear Algebra Appl., 317 (2000), 53–102 | DOI | MR | Zbl

[12] Ikramov Kh. D., “Ob odnom kriterii kvazidiagonalizuemosti veschestvennykh matrits”, ZhVMiMF, 40:1 (2000), 6–20 | MR | Zbl

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