scholarly journals The image of multilinear polynomials evaluated on 3 × 3 upper triangular matrices

2021 ◽  
Vol 29 (2) ◽  
pp. 183-186
Author(s):  
Thiago Castilho de Mello
Keyword(s):  
Matrix Algebra ◽  
Triangular Matrix ◽  
Infinite Field ◽  
Arbitrary Degree ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Triangular Matrix Algebra ◽  
Upper Triangular Matrix ◽  
Multilinear Polynomials

Abstract We describe the images of multilinear polynomials of arbitrary degree evaluated on the 3×3 upper triangular matrix algebra over an infinite field.

A NOTE ON GRADED GELFAND–KIRILLOV DIMENSION OF GRADED ALGEBRAS

2011 ◽  
Vol 10 (05) ◽  
pp. 865-889 ◽  
Author(s):  
LUCIO CENTRONE
Keyword(s):  
Triangular Matrix ◽  
Grassmann Algebra ◽  
Graded Algebras ◽  
Triangular Matrices ◽  
Infinite Dimensional ◽  
Upper Triangular Matrices ◽  
Triangular Matrix Algebra ◽  
Upper Triangular Matrix ◽  
A Finite Group ◽  
Kirillov Dimension

In this paper, we consider associative P.I. algebras over a field F of characteristic 0, graded by a finite group G. More precisely, we define the G-graded Gelfand–Kirillov dimension of a G-graded P.I. algebra. We find a basis of the relatively free graded algebras of the upper triangular matrices UTn(F) and UTn(E), with entries in F and in the infinite-dimensional Grassmann algebra, respectively. As a consequence, we compute their graded Gelfand–Kirillov dimension with respect to the natural gradings defined over these algebras. We obtain similar results for the upper triangular matrix algebra UTa, b(E) = UTa+b(E)∩Ma, b(E) with respect to its natural ℤa+b × ℤ2-grading. Finally, we compute the ℤn-graded Gelfand–Kirillov dimension of Mn(F) in some particular cases and with different methods.

Inverse Eigenvalue Problem for a Class of Upper Triangular Matrices with Linear Relation

2014 ◽  
Vol 668-669 ◽  
pp. 1068-1071
Author(s):  
Zhi Bin Li ◽  
Shuai Li
Keyword(s):  
Eigenvalue Problem ◽  
Linear Relation ◽  
Existence And Uniqueness ◽  
Triangular Matrix ◽  
Inverse Eigenvalue Problem ◽  
Triangular Matrices ◽  
Numerical Example ◽  
Upper Triangular Matrices ◽  
Inverse Eigenvalue ◽  
Upper Triangular Matrix

This paper studies on the eigenvalue[1-5] of the a class of upper triangular matrix with linear relation. It discusses the feature of existence and uniqueness of matrix via two given two characteristic pairs(λ,χ),(μ,γ) . Solutions and expressions are provided under satisfied conditions. The possibilities are exanimated by numerical example.

Graded Involutions on Upper-triangular Matrix Algebras

2009 ◽  
Vol 16 (01) ◽  
pp. 103-108 ◽  
Author(s):  
A. Valenti ◽  
M. V. Zaicev
Keyword(s):  
Abelian Group ◽  
Characteristic Zero ◽  
Finite Abelian Group ◽  
Triangular Matrix ◽  
Closed Field ◽  
Triangular Matrices ◽  
Algebraically Closed Field ◽  
Matrix Algebras ◽  
Upper Triangular Matrices ◽  
Upper Triangular Matrix

Let UTn be the algebra of n × n upper-triangular matrices over an algebraically closed field of characteristic zero. We describe all G-gradings on UTn by a finite abelian group G commuting with an involution (involution gradings).

scholarly journals A Class of Nonlinear Nonglobal Semi-Jordan Triple Derivable Mappings on Triangular Algebras

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Xiuhai Fei ◽  
Haifang Zhang
Keyword(s):  
Matrix Algebra ◽  
Triangular Matrix ◽  
Nest Algebra ◽  
Triangular Algebra ◽  
Triangular Matrix Algebra ◽  
Free Block ◽  
Upper Triangular Matrix ◽  
Triangular Algebras ◽  
Torsion Free

In this paper, we proved that each nonlinear nonglobal semi-Jordan triple derivable mapping on a 2-torsion free triangular algebra is an additive derivation. As its application, we get the similar conclusion on a nest algebra or a 2-torsion free block upper triangular matrix algebra, respectively.

Model theory of strictly upper triangular matrix rings

1980 ◽  
Vol 45 (3) ◽  
pp. 455-463 ◽  
Author(s):  
William H. Wheeler
Keyword(s):  
Model Theory ◽  
Triangular Matrix ◽  
Closed Field ◽  
Matrix Rings ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Closed Fields ◽  
Upper Triangular Matrix ◽  
Algebraically Closed Fields ◽  
Model Completion

Two questions on rings of strictly upper triangular matrices arising from B. Rose's work [5] are answered in this paper. An n × n matrix (αi, j) is strictly upper triangular if αi, j = 0 whenever i ≥ j. The ring of strictly upper triangular n × n matrices with entries from a field F will be denoted by Sn(F). Throughout this paper n will be an integer greater than 2. B. Rose [5] has shown that the complete theory of Sn(F) for an algebraically closed field F is ℵ1categorical. The first main result of this paper is that the rings Sn(F) and Sn(K) for fields F and K are isomorphic or elementarily equivalent if and only if F and K are isomorphic or elementarily equivalent, respectively (Corollary 1.6 and Theorem 2.2). This result shortens the proof of B. Rose's categoricity theorem [5, Theorem 7] by avoiding the co-stability considerations; furthermore, this result yields a proof of the converse of this categoricity theorem. The second main result is that the theory of rings of strictly upper triangular n × n matrices over algebraically closed fields is the model-completion of the theory of rings of strictly upper triangular n × n matrices over arbitrary fields (Theorem 2.5). This answers affirmatively the two conjectures at the end of [5].

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