Loading Events

An online session of the Algebra and Logic Seminar will be held on October 22, 2021 (Friday) at 2:00 p.m. (UTC +3).

A talk on

On Some Special Decompositions of Matrices over Fields and Finite Commutative Rings

will be delivered by Peter Danchev.

Abstract. In order to find a suitable expression of an arbitrary square matrix over an arbitrary field, we prove that every square matrix over an infinite field is always representable as a sum of a diagonalizable matrix and a square-zero nilpotent matrix. In addition, each 2 x 2 matrix over any field admits such a representation. We also show that, for all natural numbers n > 2, every n x n matrix over a finite field having no less than n + 1 elements also admits such a decomposition. As a consequence of these decompositions, we show that every matrix over a finite field can be expressed as the sum of a potent matrix and a square-zero nilpotent matrix. Moreover, we prove that every matrix over a finite commutative ring is always representable as a sum of a potent matrix and a square-zero nilpotent matrix, provided the Jacobson radical of the former ring has zero-square. Our main theorems substantially improve on recent results due to Abyzov et al. in Mat. Zametki (2017), Ster in Lin. Algebra & Appl. (2018), Breaz in Lin. Algebra & Appl. (2018) and Shitov in Indag. Math. (2019).

The results have been published partially in the following articles:
 
(1) P. Danchev, E. Garcia, M. G. Lozano, Decompositions of matrices into diagonalizable and square-zero matrices, Linear & Multilinear Algebra (in press 2022), published online https://doi.org/10.1080/03081087.2020.1862742 .
 
(2) P. Danchev, E. Garcia, M. G. Lozano, Decompositions of matrices into potent and square-zero matrices, submitted to a scientific journal.
 
(3) P. Danchev, E. Garcia, M. G. Lozano, On some special matrix decompositions over fields and finite commutative rings, Proceedings of the Fiftieth Spring Conference of the Union of Bulgarian Mathematicians, 95-101, 2021.

The seminar will be held via Zoom and everyone can join at:

https://us02web.zoom.us/j/85137375021?pwd=RE5QczdFTE1xL1R6MnI2b1lkcGczQT09 

Topic: Онлайн семинар на секция “Алгебра и логика”
Time: Oct 22, 2021 02:00 PM Sofia
Meeting ID: 851 3737 5021
Passcode: 035647

Algebra and Logic Department, IMI-BAS
http://www.math.bas.bg/algebra/seminarAiL/
============================== =====================