Representations of the Group of Upper Triangular Matrices

1991 ◽  
pp. 124-144
Author(s):  
Y. S. Samoilenko
Keyword(s):  
Triangular Matrices ◽  
Upper Triangular Matrices

An algorithm for decomposing a class of matrices into lower and upper triangular matrices

1990 ◽  
Vol 327 (4) ◽  
pp. 573-577
Author(s):  
Reza R. Adhami ◽  
David J. Lanteigne ◽  
Don A. Gregory
Keyword(s):  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Class Of Matrices

Applying quick exponentiation for block upper triangular matrices

2006 ◽  
Vol 183 (2) ◽  
pp. 729-737 ◽  
Author(s):  
Rafael Álvarez ◽  
Francisco Ferrández ◽  
José-Francisco Vicent ◽  
Antonio Zamora
Keyword(s):  
Triangular Matrices ◽  
Upper Triangular Matrices

Nonlinear Jordan centralizer of strictly upper triangular matrices

2019 ◽  
Vol 26 (1/2) ◽  
pp. 197-201
Author(s):  
Driss Aiat Hadj Ahmed
Keyword(s):  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Zero Characteristic

Let ℱ be a field of zero characteristic, let Nn(ℱ) denote the algebra of n×n strictly upper triangular matrices with entries in ℱ, and let f:Nn(ℱ)→Nn(ℱ) be a nonlinear Jordan centralizer of Nn(ℱ),that is, a map satisfying that f(XY+YX)=Xf(Y)+f(Y)X, for all X, Y∈Nn(ℱ). We prove that f(X)=λX+η(X) where λ∈ℱ and η is a map from Nn(ℱ) into its center 𝒵(Nn(ℱ)) satisfying that η(XY+YX)=0 for every X,Yin Nn(F).

Automorphism group of the one-divisor graph over the semigroup of upper triangular matrices

2020 ◽  
pp. 2150214
Author(s):  
Xinlei Wang ◽  
Dein Wong ◽  
Fenglei Tian
Keyword(s):  
Automorphism Group ◽  
Zero Divisor ◽  
Symmetric Groups ◽  
Directed Edge ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Zero Divisor Graph ◽  
Vertex Set ◽  
The One ◽  
Definition Of

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] a positive integer, [Formula: see text] the semigroup of all [Formula: see text] upper triangular matrices over [Formula: see text] under matrix multiplication, [Formula: see text] the group of all invertible matrices in [Formula: see text], [Formula: see text] the quotient group of [Formula: see text] by its center. The one-divisor graph of [Formula: see text], written as [Formula: see text], is defined to be a directed graph with [Formula: see text] as vertex set, and there is a directed edge from [Formula: see text] to [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text] and [Formula: see text] are, respectively, a left divisor and a right divisor of a rank one matrix in [Formula: see text]. The definition of [Formula: see text] is motivated by the definition of zero-divisor graph [Formula: see text] of [Formula: see text], which has vertex set of all nonzero zero-divisors in [Formula: see text] and there is a directed edge from a vertex [Formula: see text] to a vertex [Formula: see text] if and only if [Formula: see text], i.e. [Formula: see text]. The automorphism group of zero-divisor graph [Formula: see text] of [Formula: see text] was recently determined by Wang [A note on automorphisms of the zero-divisor graph of upper triangular matrices, Lin. Alg. Appl. 465 (2015) 214–220.]. In this paper, we characterize the automorphism group of one-divisor graph [Formula: see text] of [Formula: see text], proving that [Formula: see text], where [Formula: see text] is the automorphism group of field [Formula: see text], [Formula: see text] is a direct product of some symmetric groups. Besides, an application of automorphisms of [Formula: see text] is given in this paper.

Automorphisms of the total graph over upper triangular matrices

2019 ◽  
Vol 19 (08) ◽  
pp. 2050161
Author(s):  
Long Wang ◽  
Xianwen Fang ◽  
Fenglei Tian
Keyword(s):  
Finite Field ◽  
Total Graph ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Zero Divisors ◽  
Singular Matrices

Let [Formula: see text] be a finite field, [Formula: see text] the ring of all [Formula: see text] upper triangular matrices over [Formula: see text], [Formula: see text] the set of all zero-divisors of [Formula: see text], i.e. [Formula: see text] consists of all [Formula: see text] upper triangular singular matrices over [Formula: see text]. The total graph of [Formula: see text], denoted by [Formula: see text], is a graph with all elements of [Formula: see text] as vertices, and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we determine all automorphisms of the total graph [Formula: see text] of [Formula: see text].

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