Algebra and Logic Seminar
22 October @ 14:00 - 15:30
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 seminar will be held via Zoom and everyone can join at:
Topic: Онлайн семинар на секция “Алгебра и логика”
Time: Oct 22, 2021 02:00 PM Sofia
Meeting ID: 851 3737 5021
Algebra and Logic Department, IMI-BAS