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.

scholarly journals (Anti-)Hermitian Generalized (Anti-)Hamiltonian Solution to a System of Matrix Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Juan Yu ◽  
Qing-Wen Wang ◽  
Chang-Zhou Dong
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Optimal Approximation ◽  
Matrix Equations ◽  
Approximation Solution ◽  
Numerical Examples ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient ◽  
The Given

We mainly solve three problems. Firstly, by the decomposition of the (anti-)Hermitian generalized (anti-)Hamiltonian matrices, the necessary and sufficient conditions for the existence of and the expression for the (anti-)Hermitian generalized (anti-)Hamiltonian solutions to the system of matrix equationsAX=B,XC=Dare derived, respectively. Secondly, the optimal approximation solutionmin⁡X∈K⁡∥X^-X∥is obtained, whereKis the (anti-)Hermitian generalized (anti-)Hamiltonian solution set of the above system andX^is the given matrix. Thirdly, the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solutions are considered. In addition, algorithms about computing the least squares (anti-)Hermitian generalized (anti-)Hamiltonian solution and the corresponding numerical examples are presented.

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 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>

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].

scholarly journals Solving Optimization Problems on Hermitian Matrix Functions with Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Xiang Zhang ◽  
Shu-Wen Xiang
Keyword(s):  
Optimization Problems ◽  
Hermitian Matrix ◽  
Sufficient Conditions ◽  
Positive Definite ◽  
Matrix Functions ◽  
Matrix Equations ◽  
Necessary And Sufficient Condition ◽  
System Of Matrix Equations ◽  
The Matrix ◽  
Necessary And Sufficient

We consider the extremal inertias and ranks of the matrix expressionsf(X,Y)=A3-B3X-(B3X)*-C3YD3-(C3YD3)*, whereA3=A3*,  B3,  C3, andD3are known matrices andYandXare the solutions to the matrix equationsA1Y=C1,YB1=D1, andA2X=C2, respectively. As applications, we present necessary and sufficient condition for the previous matrix functionf(X,Y)to be positive (negative), non-negative (positive) definite or nonsingular. We also characterize the relations between the Hermitian part of the solutions of the above-mentioned matrix equations. Furthermore, we establish necessary and sufficient conditions for the solvability of the system of matrix equationsA1Y=C1,YB1=D1,A2X=C2, andB3X+(B3X)*+C3YD3+(C3YD3)*=A3, and give an expression of the general solution to the above-mentioned system when it is solvable.

scholarly journals On the HermitianR-Conjugate Solution of a System of Matrix Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Chang-Zhou Dong ◽  
Qing-Wen Wang ◽  
Yu-Ping Zhang
Keyword(s):  
Least Squares ◽  
Sufficient Conditions ◽  
Approximation Problem ◽  
Matrix Equations ◽  
Complex Matrix ◽  
Solvability Conditions ◽  
Numerical Examples ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient ◽  
Conjugate Solution

LetRbe annbynnontrivial real symmetric involution matrix, that is,R=R−1=RT≠In. Ann×ncomplex matrixAis termedR-conjugate ifA¯=RAR, whereA¯denotes the conjugate ofA. We give necessary and sufficient conditions for the existence of the HermitianR-conjugate solution to the system of complex matrix equationsAX=C and XB=Dand present an expression of the HermitianR-conjugate solution to this system when the solvability conditions are satisfied. In addition, the solution to an optimal approximation problem is obtained. Furthermore, the least squares HermitianR-conjugate solution with the least norm to this system mentioned above is considered. The representation of such solution is also derived. Finally, an algorithm and numerical examples are given.

scholarly journals On circulant codes with prescribed distances

1977 ◽  
Vol 16 (3) ◽  
pp. 361-369
Author(s):  
M. Deza ◽  
Peter Eades
Keyword(s):  
Sufficient Conditions ◽  
Difference Sets ◽  
Necessary And Sufficient Conditions ◽  
Necessary Condition ◽  
Weighing Matrices ◽  
Cyclic Difference Sets ◽  
The Matrix ◽  
Circulant Codes ◽  
Necessary And Sufficient ◽  
Square Matrix

Necessary and sufficient conditions are given for a square matrix to te the matrix of distances of a circulant code. These conditions are used to obtain some inequalities for cyclic difference sets, and a necessary condition for the existence of circulant weighing matrices.

On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences

2021 ◽  
Vol 71 (6) ◽  
pp. 1375-1400
Author(s):  
Feyzi Başar ◽  
Hadi Roopaei
Keyword(s):  
Lower Bound ◽  
Sequence Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
The Matrix ◽  
Alpha Beta ◽  
Necessary And Sufficient ◽  
Bounded Sequences

Abstract Let F denote the factorable matrix and X ∈ {ℓp , c 0, c, ℓ ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓ p (F), ℓ ∞), (ℓ p (F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.

On Solutions of Inverse Problem for Hermitian Generalized Hamiltonian Matrices

2013 ◽  
Vol 860-863 ◽  
pp. 2727-2731
Author(s):  
Kai Fu Liang ◽  
Ming Jun Li ◽  
Ze Lin Zhu
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Design Automation ◽  
Sufficient Conditions ◽  
Matrix Equations ◽  
Complex Matrix ◽  
Linear Quadratic ◽  
Real Representation ◽  
The Matrix ◽  
Necessary And Sufficient

Hamiltonian matrices have many applications to design automation and autocontrol, in particular in the linear-quadratic autocontrol problem. This paper studies the inverse problems of generalized Hamiltonian matrices for matrix equations. By real representation of complex matrix, we give the necessary and sufficient conditions for the existence of a Hermitian generalized Hamiltonian solutions to the matrix equations, and then derive the representation of the general solutions.

Inverse problem about two-spectra for finite Jacobi matrices with zero diagonal

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Adil Huseynov
Keyword(s):  
Inverse Problem ◽  
Finite Order ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Jacobi Matrices ◽  
The Matrix ◽  
Explicit Procedure ◽  
Necessary And Sufficient

AbstractThe necessary and sufficient conditions for solvability of the inverse problem about two-spectra for finite order real Jacobi matrices with zero-diagonal elements are established. An explicit procedure of reconstruction of the matrix from the two-spectra is given.

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