Solvability conditions for mixed sylvester equations in rings

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5539-5543 ◽  
Author(s):  
Huaxi Chen ◽  
Long Wang ◽  
Qian Wang
Keyword(s):  
General Solution ◽  
Regular Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
Solvability Conditions ◽  
Sylvester Matrix ◽  
Necessary And Sufficient ◽  
Sylvester Matrix Equations ◽  
Algebraic Technique

This paper has been motivated byWang and He [Q.W.Wang and Z.H. He, Solvability conditions and general solution for mixed Sylvester equations. Automatica, 49 (2013) 2713-2719] in which the authors consider some solvability conditions for mixed Sylvester matrix equations. The paper also considers the same problem in the setting of a regular ring. Using the purely algebraic technique, we present some necessary and sufficient conditions for the solvability to mixed Sylvester equations in rings.

On the Centro-symmetric Solution of a System of Matrix Equations over a Regular Ring with Identity

2007 ◽  
Vol 14 (04) ◽  
pp. 555-570 ◽  
Author(s):  
Qingwen Wang ◽  
Haixia Chang ◽  
Chunyan Lin
Keyword(s):  
General Solution ◽  
Explicit Expression ◽  
Symmetric Solution ◽  
Regular Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient

In this paper, we find the centro-symmetric solution of a system of matrix equations over an arbitrary regular ring [Formula: see text] with identity. We first derive some necessary and sufficient conditions for the existence and an explicit expression of the general solution of the system of matrix equations A1X1 = C1, A2X1 = C2, A3X2 = C3, A4X2 = C4 and A5X1B5 + A6X2B6 = C5 over [Formula: see text]. By using the above results, we establish two criteria for the existence and the representation of the general centro-symmetric solution of the system of matrix equations AaX = Ca, AbX = Cb and AcXBc = Cc over the ring [Formula: see text].

A System of Periodic Discrete-time Coupled Sylvester Quaternion Matrix Equations

2017 ◽  
Vol 24 (01) ◽  
pp. 169-180 ◽  
Author(s):  
Zhuoheng He ◽  
Qingwen Wang
Keyword(s):  
General Solution ◽  
Discrete Time ◽  
Quaternion Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Generalized Inverses ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Necessary And Sufficient ◽  
Sylvester Matrix Equations

We in this paper derive necessary and sufficient conditions for the system of the periodic discrete-time coupled Sylvester matrix equations [Formula: see text] over the quaternion algebra to be consistent in terms of ranks and generalized inverses of the coefficient matrices. We also give an expression of the general solution to the system when it is solvable. The findings of this paper generalize some known results in the literature.

A System of Matrix Equations over the Quaternion Algebra with Applications

2017 ◽  
Vol 24 (02) ◽  
pp. 233-253 ◽  
Author(s):  
Xiangrong Nie ◽  
Qingwen Wang ◽  
Yang Zhang
Keyword(s):  
General Solution ◽  
Quaternion Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
Electronic Networks ◽  
Consistency Conditions ◽  
Numerical Example ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient

We in this paper give necessary and sufficient conditions for the existence of the general solution to the system of matrix equations [Formula: see text] and [Formula: see text] over the quaternion algebra ℍ, and present an expression of the general solution to this system when it is solvable. Using the results, we give some necessary and sufficient conditions for the system of matrix equations [Formula: see text] over ℍ to have a reducible solution as well as the representation of such solution to the system when the consistency conditions are met. A numerical example is also given to illustrate our results. As another application, we give the necessary and sufficient conditions for two associated electronic networks to have the same branch current and branch voltage and give the expressions of the same branch current and branch voltage when the conditions are satisfied.

scholarly journals Least-Norm of the General Solution to Some System of Quaternion Matrix Equations and Its Determinantal Representations

2019 ◽  
Vol 2019 ◽  
pp. 1-18
Author(s):  
Abdur Rehman ◽  
Ivan Kyrchei ◽  
Muhammad Akram ◽  
Ilyas Ali ◽  
Abdul Shakoor
Keyword(s):  
General Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Skew Field ◽  
Numerical Examples ◽  
Cramer’S Rule ◽  
Necessary And Sufficient

We constitute some necessary and sufficient conditions for the system A1X1=C1, X1B1=C2, A2X2=C3, X2B2=C4, A3X1B3+A4X2B4=Cc, to have a solution over the quaternion skew field in this paper. A novel expression of general solution to this system is also established when it has a solution. The least norm of the solution to this system is also researched in this article. Some former consequences can be regarded as particular cases of this article. Finally, we give determinantal representations (analogs of Cramer’s rule) of the least norm solution to the system using row-column noncommutative determinants. An algorithm and numerical examples are given to elaborate our results.

The Common Solution of Some Matrix Equations

2016 ◽  
Vol 23 (01) ◽  
pp. 71-81 ◽  
Author(s):  
Li Wang ◽  
Qingwen Wang ◽  
Zhuoheng He
Keyword(s):  
General Solution ◽  
Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Existence Of A Solution ◽  
Solvability Conditions ◽  
Common Solution ◽  
The Common ◽  
Necessary And Sufficient ◽  
Linear Matrix Equations

In this paper we investigate the system of linear matrix equations A1X=C1, YB2=C2, A3XB3=C3, A4YB4=C4, BX+YC=A. We present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied.

Regular, Commutative, Maximal Semigroups of Quotients

1975 ◽  
Vol 18 (1) ◽  
pp. 99-104 ◽  
Author(s):  
Jurgen Rompke
Keyword(s):  
Commutative Semigroup ◽  
Regular Ring ◽  
Sufficient Conditions ◽  
Commutative Rings ◽  
Necessary And Sufficient Conditions ◽  
Natural Question ◽  
Commutative Case ◽  
Ring Of Quotients ◽  
Singular Ideal ◽  
Necessary And Sufficient

A well-known theorem which goes back to R. E. Johnson [4], asserts that if R is a ring then Q(R), its maximal ring of quotients is regular (in the sense of v. Neumann) if and only if the singular ideal of R vanishes. In the theory of semigroups a natural question is therefore the following: Do there exist properties which characterize those semigroups whose maximal semigroups of quotients are regular? Partial answers to this question have been given in [3], [7] and [8]. In this paper we completely solve the commutative case, i.e. we give necessary and sufficient conditions for a commutative semigroup S in order that Q(S), the maximal semigroup of quotients, is regular. These conditions reflect very closely the property of being semiprime, which in the theory of commutative rings characterizes those rings which have a regular ring of quotients.

Maximal and Minimal Ranks of the Common Solution of Some Linear Matrix Equations over an Arbitrary Division Ring with Applications

2009 ◽  
Vol 16 (02) ◽  
pp. 293-308 ◽  
Author(s):  
Qingwen Wang ◽  
Guangjing Song ◽  
Xin Liu
Keyword(s):  
Division Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix Equations ◽  
Common Solution ◽  
Special Cases ◽  
The Common ◽  
Necessary And Sufficient ◽  
Linear Matrix Equations

We establish the formulas of the maximal and minimal ranks of the common solution of certain linear matrix equations A1X = C1, XB2 = C2, A3XB3 = C3 and A4XB4 = C4 over an arbitrary division ring. Corresponding results in some special cases are given. As an application, necessary and sufficient conditions for the invariance of the rank of the common solution mentioned above are presented. Some previously known results can be regarded as special cases of our results.

scholarly journals Pure PSVD approach to Sylvester-type quaternion matrix equations

2019 ◽  
Vol 35 ◽  
pp. 266-284 ◽  
Author(s):  
Zhuo-Heng He
Keyword(s):  
General Solution ◽  
Sufficient Conditions ◽  
Singular Value ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Existence Of A Solution ◽  
Quaternion Matrices ◽  
Pure Product ◽  
Value Decomposition ◽  
Necessary And Sufficient

In this paper, the pure product singular value decomposition (PSVD) for four quaternion matrices is given. The system of coupled Sylvester-type quaternion matrix equations with five unknowns $X_{i}A_{i}-B_{i}X_{i+1}=C_{i}$ is considered by using the PSVD approach, where $A_{i},B_{i},$ and $C_{i}$ are given quaternion matrices of compatible sizes $(i=1,2,3,4)$. Some necessary and sufficient conditions for the existence of a solution to this system are derived. Moreover, the general solution to this system is presented when it is solvable.

scholarly journals The common Re-nonnegative definite and Re-positive definite solutions to the matrix equations $ A_1XA_1^\ast = C_1 $ and $ A_2XA_2^\ast = C_2 $

2021 ◽  
Vol 7 (1) ◽  
pp. 384-397
Author(s):  
Yinlan Chen ◽  
◽  
Lina Liu
Keyword(s):  
Sufficient Conditions ◽  
Positive Definite ◽  
Necessary And Sufficient Conditions ◽  
Linear Matrix ◽  
Matrix Equations ◽  
The Matrix ◽  
The Common ◽  
Necessary And Sufficient ◽  
Nonnegative Definite ◽  
Linear Matrix Equations

<abstract><p>In this paper, we consider the common Re-nonnegative definite (Re-nnd) and Re-positive definite (Re-pd) solutions to a pair of linear matrix equations $ A_1XA_1^\ast = C_1, \ A_2XA_2^\ast = C_2 $ and present some necessary and sufficient conditions for their solvability as well as the explicit expressions for the general common Re-nnd and Re-pd solutions when the consistent conditions are satisfied.</p></abstract>

scholarly journals Constrained Solutions of a System of Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
Qing-Wen Wang ◽  
Juan Yu
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
System Of Matrix Equations ◽  
The Matrix ◽  
Necessary And Sufficient ◽  
Constrained Solutions

We derive the necessary and sufficient conditions of and the expressions for the orthogonal solutions, the symmetric orthogonal solutions, and the skew-symmetric orthogonal solutions of the system of matrix equationsAX=BandXC=D, respectively. When the matrix equations are not consistent, the least squares symmetric orthogonal solutions and the least squares skew-symmetric orthogonal solutions are respectively given. As an auxiliary, an algorithm is provided to compute the least squares symmetric orthogonal solutions, and meanwhile an example is presented to show that it is reasonable.

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