scholarly journals The algebra generated by nilpotent elements in a matrix centralizer

2021 ◽  
pp. 1-8
Author(s):  
Ralph John De la Cruz ◽  
Eloise Misa
Keyword(s):  
Canonical Form ◽  
Nilpotent Elements ◽  
Square Matrix

For an arbitrary square matrix $S$, denote by $C(S)$ the centralizer of $S$, and by $C(S)_N$ the set of all nilpotent elements in $C(S)$. In this paper, we use the Weyr canonical form to study the subalgebra of $C(S)$ generated by $C(S)_N$. We determine conditions on $S$ such that $C(S)_N$ is a subalgebra of $C(S)$. We also determine conditions on $S$ such that the subalgebra generated by $C(S)_N$ is $C(S).$

Further Solutions of a Yang-Baxter-like Matrix Equation

2013 ◽  
Vol 3 (4) ◽  
pp. 352-362 ◽  
Author(s):  
Jiu Ding ◽  
Chenhua Zhang ◽  
Noah H. Rhee
Keyword(s):  
Canonical Form ◽  
Infinite Number ◽  
Matrix Equation ◽  
Number Of Solutions ◽  
Jordan Canonical Form ◽  
The Matrix ◽  
Square Matrix

AbstractThe Yang-Baxter-like matrix equation AXA = XAX is reconsidered, and an infinite number of solutions that commute with any given complex square matrix A are found. Our results here are based on the fact that the matrix A can be replaced with its Jordan canonical form. We also discuss the explicit structure of the solutions obtained.

scholarly journals On the Canonical Form of a Rational Integral Function of a Matrix

1932 ◽  
Vol 3 (2) ◽  
pp. 135-143 ◽  
Author(s):  
D. E. Rutherford
Keyword(s):  
Canonical Form ◽  
Integral Function ◽  
The Other ◽  
Singular Matrix ◽  
The Matrix ◽  
Image Position ◽  
Rational Integral ◽  
Square Matrix

It is well known that the square matrix, of rank n−k + 1,which we shall denote by B where any element to the left of, or below the nonzero diagonal b1, k, b2, k + 1, . …, bn−k + 1, n is zero, can be resolved into factors Z−1DZ; where D is a square matrix of order n having the elements d1, k, d2, k + 1, . …, dn−k + 1, n all unity and all the other elements zero, and where Z is a non-singular matrix. In this paper we shall show in a particular case that this is so, and in the case in question we shall exhibit the matrix Z explicitly. Application of this is made to find the classical canonical form of a rational integral function of a square matrix A.

Canonical Form of Square Matrix

2011 ◽  
Vol 114 (1106) ◽  
pp. 53-61
Author(s):  
Akihiko SHIMIZU
Keyword(s):  
Canonical Form ◽  
Square Matrix

scholarly journals A remark about canonical forms

1956 ◽  
Vol 40 ◽  
pp. 15-15
Author(s):  
Hazel Perfect
Keyword(s):  
Canonical Form ◽  
Canonical Forms ◽  
Nilpotent Matrix ◽  
Square Matrix

A comparison of the rational and classical canonical forms of a square matrix reveals that for a nilpotent matrix the two are identical. In this note I describe how we may utilise this fact in solving the problem of reducing a given matrix to classical canonical form. I believe that the point which I try to make in what follows is one which is not always explicitly remarked upon in the literature, and it has therefore seemed to me to be worth while to stress it here.

A Jordan canonical form for nilpotent elements in an arbitrary ring

2019 ◽  
Vol 581 ◽  
pp. 324-335
Author(s):  
Esther García ◽  
Miguel Gómez Lozano ◽  
Rubén Muñoz Alcázar ◽  
Guillermo Vera de Salas
Keyword(s):  
Canonical Form ◽  
Jordan Canonical Form ◽  
Arbitrary Ring ◽  
Nilpotent Elements

On quasi-radical of near-ring of polynomials

2019 ◽  
Vol 56 (2) ◽  
pp. 252-259
Author(s):  
Ebrahim Hashemi ◽  
Fatemeh Shokuhifar ◽  
Abdollah Alhevaz
Keyword(s):  
Commutative Ring ◽  
Nilpotent Elements ◽  
Ring Of Polynomials

Abstract The intersection of all maximal right ideals of a near-ring N is called the quasi-radical of N. In this paper, first we show that the quasi-radical of the zero-symmetric near-ring of polynomials R0[x] equals to the set of all nilpotent elements of R0[x], when R is a commutative ring with Nil (R)2 = 0. Then we show that the quasi-radical of R0[x] is a subset of the intersection of all maximal left ideals of R0[x]. Also, we give an example to show that for some commutative ring R the quasi-radical of R0[x] coincides with the intersection of all maximal left ideals of R0[x]. Moreover, we prove that the quasi-radical of R0[x] is the greatest quasi-regular (right) ideal of it.

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