| Full text | |
| Author(s): |
Bondarenko, Vitalij M.
[1]
;
Futorny, Vyacheslav
[2]
;
Petravchuk, Anatolii P.
[3]
;
Sergeichuk, Vladimir V.
[1]
Total Authors: 4
|
| Affiliation: | [1] Inst Math, Tereshchenkivska 3, Kiev - Ukraine
[2] Univ Sao Paulo, Dept Math, Sao Paulo - Brazil
[3] Taras Shevchenko Univ, Fac Mech & Math, Kiev - Ukraine
Total Affiliations: 3
|
| Document type: | Journal article |
| Source: | Linear Algebra and its Applications; v. 612, p. 188-205, MAR 1 2021. |
| Web of Science Citations: | 0 |
| Abstract | |
I.M. Gelfand and V.A. Ponomarev (1969) proved that the problem of classifying pairs (A, B) of commuting nilpotent operators on a vector space contains the problem of classifying an arbitrary t-tuple of linear operators. Moreover, it contains the problem of classifying representations of an arbitrary quiver and an arbitrary finite-dimensional algebra, and so it is considered as hopeless. If (A, B) is such a pair, then Ker A boolean AND Ker B not equal 0. We give a simple normal form (A(nor),B-nor) of the matrices of (A, B) if Ker A boolean AND Ker B is one-dimensional. We do not know whether it is canonical; i.e., whether (A(nor), B-nor) is uniquely determined by (A , B). We prove its uniqueness only if the Jordan canonical form of A is a direct sum of Jordan blocks of the same size and the field is of zero characteristic. The matrix A(nor) is the Weyr canonical form of A, and B-nor commutes with A(nor). In order to describe the structure of (A(nor), B-nor), we describe explicitly all matrices commuting with a given Weyr matrix. (C) 2020 Elsevier Inc. All rights reserved. (AU) | |
| FAPESP's process: | 18/24089-4 - Classification problems for matrices, matrix spaces and tensors |
| Grantee: | Vyacheslav Futorny |
| Support Opportunities: | Research Grants - Visiting Researcher Grant - International |
| FAPESP's process: | 18/23690-6 - Structures, representations, and applications of algebraic systems |
| Grantee: | Ivan Chestakov |
| Support Opportunities: | Research Projects - Thematic Grants |