scholarly journals Solutions to the System of Operator EquationsA1X=C1,XB2=C2, andA3XB3=C3on HilbertC*-Modules

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaochun Fang ◽  
Enran Hou ◽  
Ge Dong
Keyword(s):  
General Solution ◽  
Positive Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Operator Equations ◽  
Existence Of A Solution ◽  
Necessary And Sufficient

We study the solvability of the system of the adjointable operator equationsA1X=C1,XB2=C2, andA3XB3=C3over HilbertC*-modules. We give necessary and sufficient conditions for the existence of a solution and a positive solution of the system. We also derive representations for a general solution and a positive solution to this system. The above results generalize some recent results concerning the equations for operators with closed ranges.

scholarly journals A NOTE ON SOME SYSTEMS OF GENERALIZED SYLVESTER EQUATIONS

2021 ◽  
pp. 449
Author(s):  
Jovana Nikolov Radenković
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Operator Equations ◽  
Existence Of A Solution ◽  
Two Systems ◽  
Necessary And Sufficient

In this paper, we study two systems of generalized Sylvester operator equations. We derive necessary and sufficient conditions for the existence of a solution and provide the general form of a solution. We extend some recent results to more general settings.

scholarly journals Positive solutions of the system of operator equations $A_1X=C_1,XA_2=C_2, A_3XA^*_3=C_3, A_4XA^*_4=C_4$ in Hilbert $C^*$-modules

2018 ◽  
Vol 34 ◽  
pp. 381-388 ◽  
Author(s):  
Rasoul Eskandari ◽  
Xiaochun Fang ◽  
Mohammad Sal Moslehian ◽  
Qingxiang Xu
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Operator Equations ◽  
Operator System ◽  
Necessary And Sufficient ◽  
System Of Operator Equations ◽  
Published Result

Necessary and sufficient conditions are given for the operator system $A_1X=C_1$, $XA_2=C_2$, $A_3XA^*_3=C_3$, and $A_4XA^*_4=C_4$ to have a common positive solution, where $A_i$'s and $C_i$'s are adjointable operators on Hilbert $C^*$-modules. This corrects a published result by removing some gaps in its proof. Finally, a technical example is given to show that the proposed investigation in the setting of Hilbert $C^*$-modules is different from that of Hilbert spaces.

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 Eigenvalue assignment in fractional descriptor discrete-time linear systems

2017 ◽  
Vol 27 (1) ◽  
pp. 119-128
Author(s):  
Tadeusz Kaczorek ◽  
Kamil Borawski
Keyword(s):  
Linear Systems ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Existence Of A Solution ◽  
Numerical Example ◽  
Necessary And Sufficient ◽  
Time Linear ◽  
Eigenvalue Assignment

Abstract The problem of eigenvalue assignment in fractional descriptor discrete-time linear systems is considered. Necessary and sufficient conditions for the existence of a solution to the problem are established. A procedure for computation of the gain matrices is given and illustrated by a numerical example.

scholarly journals On the solvability of systems of linear equations on commutative semigroup

2010 ◽  
Vol 43 (4) ◽  
Author(s):  
Nguyen Thi Thu Huyen ◽  
Nguyen Minh Tuan
Keyword(s):  
Linear Operator ◽  
Linear Space ◽  
Commutative Semigroup ◽  
Linear Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Operator Equations ◽  
Systems Of Linear Equations ◽  
Necessary And Sufficient ◽  
Linear Operator Equations

AbstractThis paper deals with the solvability of systems of linear operator equations in a linear space. Namely, the paper provides necessary and sufficient conditions for the operators under which certain kinds of systems of operator equations are solvable.

scholarly journals Necessary and sufficient conditions for recurrence and transience of Markov chains, in terms of inequalities

1978 ◽  
Vol 15 (4) ◽  
pp. 848-851 ◽  
Author(s):  
Jean-François Mertens ◽  
Ester Samuel-Cahn ◽  
Shmuel Zamir
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Bounded Solution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Existence Of A Solution ◽  
Necessary And Sufficient ◽  
Irreducible Markov Chain

For an aperiodic, irreducible Markov chain with the non-negative integers as state space it is shown that the existence of a solution to in which yi → ∞is necessary and sufficient for recurrence, and the existence of a bounded solution to the same inequalities, with yk < yo, · · ·, yN–1 for some k ≧ N, is necessary and sufficient for transience.

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.

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