scholarly journals The Number of Solutions of $X^2=0$ in Triangular Matrices Over $GF(q)$

10.37236/1226 ◽  
1995 ◽  
Vol 3 (1) ◽  
Author(s):  
Shalosh B. Ekhad ◽  
Doron Zeilberger
Keyword(s):  
Explicit Formula ◽  
Number Of Solutions ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Zero Matrix

We prove an explicit formula for the number of $n \times n$ upper triangular matrices, over $GF(q)$, whose square is the zero matrix. This formula was recently conjectured by Sasha Kirillov and Anna Melnikov.

scholarly journals Counting self-dual interval orders

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Vít Jelínek
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Interval Orders ◽  
Maximal Elements ◽  
Triangular Matrices ◽  
Bijective Correspondence ◽  
Upper Triangular Matrices ◽  
Positive Entry ◽  
International Audience ◽  
Natural Statistics

International audience In this paper, we first derive an explicit formula for the generating function that counts unlabeled interval orders (a.k.a. (2+2)-free posets) with respect to several natural statistics, including their size, magnitude, and the number of minimal and maximal elements. In the second part of the paper, we derive a generating function for the number of self-dual unlabeled interval orders, with respect to the same statistics. Our method is based on a bijective correspondence between interval orders and upper-triangular matrices in which each row and column has a positive entry. Dans cet article, on obtient une expression explicite pour la fonction génératrice du nombre des ensembles partiellement ordonnés (posets) qui évitent le motif (2+2). La fonction compte ces ensembles par rapport à plusieurs statistiques naturelles, incluant le nombre d'éléments, le nombre de niveaux, et le nombre d'éléments minimaux et maximaux. Dans la deuxième partie, on obtient une expression similaire pour la fonction génératrice des posets autoduaux évitant le motif (2+2). On obtient ces résultats à l'aide d'une bijection entre les posets évitant (2+2) et les matrices triangulaires supérieures dont chaque ligne et chaque colonne contient un élément positif.

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.

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