scholarly journals A note on commuting additive maps on rank k symmetric matrices

2021 ◽  
Vol 37 ◽  
pp. 734-746
Author(s):  
Wai Leong Chooi ◽  
Yean Nee Tan
Keyword(s):  
Linear Space ◽  
Symmetric Matrices ◽  
Identity Matrix ◽  
Additive Map ◽  
Additive Maps

Let $n\geq 2$ and $1<k\leq n$ be integers. Let $S_n(\mathbb{F})$ be the linear space of $n\times n$ symmetric matrices over a field $\mathbb{F}$ of characteristic not two. In this note, we prove that an additive map $\psi:S_n(\mathbb{F})\rightarrow S_n(\mathbb{F})$ satisfies $\psi(A)A=A\psi(A)$ for all rank $k$ matrices $A\in S_n(\mathbb{F})$ if and only if there exists a scalar $\lambda\in \mathbb{F}$ and an additive map $\mu:S_n(\mathbb{F})\rightarrow \mathbb{F}$ such that\[\psi(A)=\lambda A+\mu(A)I_n,\]for all $A\in S_n(\mathbb{F})$, where $I_n$ is the identity matrix. Examples showing the indispensability of assumptions on the integer $k>1$ and the underlying field $\mathbb{F}$ of characteristic not two are included.

Additive Maps on Units of Rings

2018 ◽  
Vol 61 (1) ◽  
pp. 130-141
Author(s):  
Tamer Košan ◽  
Serap Sahinkaya ◽  
Yiqiang Zhou
Keyword(s):  
Recent Result ◽  
Direct Product ◽  
Homomorphic Image ◽  
Semilocal Ring ◽  
Additive Map ◽  
Exchange Rings ◽  
Additive Maps

AbstractLet R be a ring. A map f: R → R is additive if f(a + b) = f(a) + f(b) for all elements a and b of R. Here, a map f: R → R is called unit-additive if f(u + v) = f(u) + f(v) for all units u and v of R. Motivated by a recent result of Xu, Pei and Yi showing that, for any field F, every unit-additive map of (F) is additive for all n ≥ z, this paper is about the question of when every unit-additivemap of a ring is additive. It is proved that every unit-additivemap of a semilocal ring R is additive if and only if either R has no homomorphic image isomorphic to or R/J(R) ≅ with 2 = 0 in R. Consequently, for any semilocal ring R, every unit-additive map of (R) is additive for all n ≥ 2. These results are further extended to rings R such that R/J(R) is a direct product of exchange rings with primitive factors Artinian. A unit-additive map f of a ring R is called unithomomorphic if f(uv) = f(u)f(v) for all units u, v of R. As an application, the question of when every unit-homomorphic map of a ring is an endomorphism is addressed.

scholarly journals Inconsistency evaluation in pairwise comparison using norm-based distances

2020 ◽  
Vol 43 (2) ◽  
pp. 657-672 ◽  
Author(s):  
Michele Fedrizzi ◽  
Nino Civolani ◽  
Andrew Critch
Keyword(s):  
Linear Space ◽  
General Framework ◽  
Pairwise Comparison ◽  
Equivalence Classes ◽  
Preference Aggregation ◽  
Symmetric Matrices ◽  
Pairwise Comparison Matrix ◽  
Group Preference ◽  
Common Value ◽  
Inconsistency Index

AbstractThis paper studies the properties of an inconsistency index of a pairwise comparison matrix under the assumption that the index is defined as a norm-induced distance from the nearest consistent matrix. Under additive representation of preferences, it is proved that an inconsistency index defined in this way is a seminorm in the linear space of skew-symmetric matrices and several relevant properties hold. In particular, this linear space can be partitioned into equivalence classes, where each class is an affine subspace and all the matrices in the same class share a common value of the inconsistency index. The paper extends in a more general framework some results due, respectively, to Crawford and to Barzilai. It is also proved that norm-based inconsistency indices satisfy a set of six characterizing properties previously introduced, as well as an upper bound property for group preference aggregation.

Additive maps having a generalized derivation expansion

2015 ◽  
Vol 14 (04) ◽  
pp. 1550048 ◽  
Author(s):  
Tsiu-Kwen Lee
Keyword(s):  
Central Simple Algebra ◽  
Simple Algebra ◽  
Generalized Derivation ◽  
Elementary Operator ◽  
Additive Map ◽  
Finite Dimensional ◽  
Left And Right ◽  
Central Simple ◽  
Skolem Theorem ◽  
Additive Maps

Let R be a prime ring with extended centroid C. We prove that an additive map from R into RC + C can be characterized in terms of left and right b-generalized derivations if it has a generalized derivation expansion. As a consequence, a generalization of the Noether–Skolem theorem is proved among other things: A linear map from a finite-dimensional central simple algebra into itself is an elementary operator if it has a generalized derivation expansion.

scholarly journals Additive Maps of Rank k Bivectors

2021 ◽  
Vol 36 (36) ◽  
pp. 847-856
Author(s):  
Wai Leong Chooi ◽  
Kiam Heong Kwa
Keyword(s):  
Linear Spaces ◽  
Exterior Power ◽  
Additive Map ◽  
Additive Maps

Let ${\cal U}$ and ${\cal V}$ be linear spaces over fields $\mathbb{F}$ and $\mathbb{K}$, respectively, such that Dim$\,{\cal U}=n\geqslant 2$ and $\left|\mathbb{F}\right|\geqslant 3$. Let $\bigwedge^2{\cal U}$ be the second exterior power of ${\cal U}$. Fixing an even integer $k$ satisfying $\frac{n-1}{2}\leqslant k\leqslant n$, it is shown that a map $\psi:\bigwedge^2{\cal U}\rightarrow\bigwedge^2{\cal V}$ satisfies $\psi(u+v)=\psi(u)+\psi(v)$ for all rank $k$ bivectors $u,v\in\bigwedge^2{\cal U}$ if and only if $\psi$ is an additive map. Examples showing the indispensability of the assumption on $k$ are given.

scholarly journals Ninety-Six Distinct Real Matrices for Representing a Quaternion Number

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
W. E. Ahmed
Keyword(s):  
Symmetric Matrices ◽  
Identity Matrix ◽  
Real Matrices

In this paper, we investigate on the number of all possible real matrices representing a quaternion number as three 4 × 4 skew-symmetric matrices plus the identity matrix of order 4, and how to determine these matrices. We establish that there are 96 distinct real matrices having this property, and by matrix row operations, we obtain these matrices.

Some Convexity Features Associated with Unitary Orbits

2003 ◽  
Vol 55 (1) ◽  
pp. 91-111 ◽  
Author(s):  
Man-Duen Choi ◽  
Chi-Kwong Li ◽  
Yiu-Tung Poon
Keyword(s):  
Convex Hull ◽  
Linear Space ◽  
Spectral Properties ◽  
Numerical Range ◽  
Symmetric Matrices ◽  
Hermitian Matrices ◽  
Linear Transformations ◽  
Unitary Similarity ◽  
Similarity Orbit ◽  
Real Linear

AbstractLet be the real linear space of n × n complex Hermitian matrices. The unitary (similarity) orbit of C ∈ is the collection of all matrices unitarily similar to C. We characterize those C ∈ such that every matrix in the convex hull of can be written as the average of two matrices in . The result is used to study spectral properties of submatrices of matrices in , the convexity of images of under linear transformations, and some related questions concerning the joint C-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.

scholarly journals On additive maps of prime rings

1995 ◽  
Vol 51 (3) ◽  
pp. 377-381 ◽  
Author(s):  
Matej Brešar ◽  
Bojan Hvala
Keyword(s):  
Prime Ring ◽  
Extended Centroid ◽  
Prime Rings ◽  
Additive Map ◽  
Additive Maps

Let R be a prime ring of characteristic not 2, C be the extended centroid of R, and f: R → R be an additive map. Suppose that [f(x), x2] = 0 for all x ∈ R. Then there exist λ ∈ C and an additive map ζ: R → C such that f(x) = λx + ζ(x) for all x ∈ R. In particular, if f(x)2 = x2 for all x ∈ R, then ζ = 0 and either λ = 1 or λ= -1.

scholarly journals m-commuting Additive Maps on Upper Triangular Matrices Rings

2019 ◽  
pp. 505-514
Author(s):  
Driss Aiat Hadj Ahmed
Keyword(s):  
Commutative Ring ◽  
Matrix Ring ◽  
Triangular Matrix ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Additive Map ◽  
Triangular Matrix Ring ◽  
Upper Triangular Matrix ◽  
Additive Maps ◽  
Upper Triangular Matrix Ring

Let $T_{n}(R)$ be the upper triangular matrix ring over a unital commutative ring whose characteristic is not a divisor of $m$. Suppose that $f:T_{n}(R)\rightarrow T_{n}(R)$ is an additive map such that $X^{m}f(X)=f(X)X^{m}$ for all $x \in T_{n}(R),$ where $m\geq 1$ is an integer. We consider the problem of describing the form of the map $X \rightarrow f(X)$.

scholarly journals Euclidean Jordan algebras and some conditions over the spectra of a strongly regular graph

4open ◽  
2019 ◽  
Vol 2 ◽  
pp. 21
Author(s):  
Luís Vieira
Keyword(s):  
Regular Graph ◽  
Strongly Regular Graph ◽  
Euclidean Jordan Algebra ◽  
Jordan Algebras ◽  
Inner Product ◽  
Symmetric Matrices ◽  
Identity Matrix ◽  
Jordan Product ◽  
Euclidean Jordan Algebras ◽  
Strongly Regular

Let G be a primitive strongly regular graph G such that the regularity is less than half of the order of G and A its matrix of adjacency, and let 𝒜 be the real Euclidean Jordan algebra of real symmetric matrices of order n spanned by the identity matrix of order n and the natural powers of A with the usual Jordan product of two symmetric matrices of order n and with the inner product of two matrices being the trace of their Jordan product. Next the spectra of two Hadamard series of 𝒜 associated to A2 is analysed to establish some conditions over the spectra and over the parameters of G.

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