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Qing-wen Wang - One of the best experts on this subject based on the ideXlab platform.
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Consistency of Quaternion Matrix Equations $AX^{\star}-XB=C$ and $X-AX^\star B=C$
The Electronic Journal of Linear Algebra, 2019Co-Authors: Xin Liu, Qing-wen Wang, Yang ZhangAbstract:For a given ordered units triple $\{q_1, q_2, q_3\}$, the solutions to the quaternion matrix equations $AX^{\star}-XB=C$ and $X-AX^{\star}B=C$, $X^{\star} \in \{ X , X^{\eta} , X^* , X^{\eta*}\}$, where $X^*$ is the Conjugate Transpose of $X$, $X^{\eta}=-\eta X \eta$ and $X^{\eta*}=-\eta X^* \eta$, $\eta \in \{q_1, q_2, q_3\}$, are discussed. Some new real representations of quaternion matrices are used, which enable one to convert $\eta$-Conjugate (Transpose) matrix equations into some real matrix equations. By using this idea, conditions for the existence and uniqueness of solutions to the above quaternion matrix equations are derived. Also, methods to construct the solutions from some related real matrix equations are presented.
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The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation
Mathematical Problems in Engineering, 2017Co-Authors: Xin Liu, Qing-wen WangAbstract:For a given generalized reflection matrix , that is, , , where is the Conjugate Transpose matrix of , a matrix is called a Hermitian (anti)reflexive matrix with respect to if and By using the Kronecker product, we derive the explicit expression of least squares Hermitian (anti)reflexive solution with the least norm to matrix equation over complex field.
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The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation AXB=C
Hindawi Limited, 2017Co-Authors: Xin Liu, Qing-wen WangAbstract:For a given generalized reflection matrix J, that is, JH=J, J2=I, where JH is the Conjugate Transpose matrix of J, a matrix A∈Cn×n is called a Hermitian (anti)reflexive matrix with respect to J if AH=A and A=±JAJ. By using the Kronecker product, we derive the explicit expression of least squares Hermitian (anti)reflexive solution with the least norm to matrix equation AXB=C over complex field
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The -Reflexive Solution to System of Matrix Equations ,
Mathematical Problems in Engineering, 2015Co-Authors: Chang-zhou Dong, Qing-wen WangAbstract:Let and be Hermitian and -potent matrices; that is, and where stands for the Conjugate Transpose of a matrix. A matrix is called -reflexive (antireflexive) if . In this paper, the system of matrix equations and subject to -reflexive and antireflexive constraints is studied by converting into two simpler cases: and We give the solvability conditions and the general solution to this system; in addition, the least squares solution is derived; finally, the associated optimal approximation problem for a given matrix is considered.
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The {P,Q,k+1}-Reflexive Solution to System of Matrix Equations AX=C, XB=D
Hindawi Limited, 2015Co-Authors: Chang-zhou Dong, Qing-wen WangAbstract:Let P∈Cm×m and Q∈Cn×n be Hermitian and {k+1}-potent matrices; that is, Pk+1=P=P⁎ and Qk+1=Q=Q⁎, where ·⁎ stands for the Conjugate Transpose of a matrix. A matrix X∈Cm×n is called {P,Q,k+1}-reflexive (antireflexive) if PXQ=X (PXQ=-X). In this paper, the system of matrix equations AX=C and XB=D subject to {P,Q,k+1}-reflexive and antireflexive constraints is studied by converting into two simpler cases: k=1 and k=2. We give the solvability conditions and the general solution to this system; in addition, the least squares solution is derived; finally, the associated optimal approximation problem for a given matrix is considered
Xin Liu - One of the best experts on this subject based on the ideXlab platform.
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\(\eta \)-Hermitian Solution to a System of Quaternion Matrix Equations
Bulletin of the Malaysian Mathematical Sciences Society, 2020Co-Authors: Xin LiuAbstract:For \(\eta \in \{{\mathbf {i}},{\mathbf {j}},{\mathbf {k}}\}\), a real quaternion matrix A is said to be \(\eta \)-Hermitian if \(A=A^{\eta *},\) where \(A^{\eta *}=-\eta A^{*}\eta \), and \(A^{*}\) stands for the Conjugate Transpose of A. In this paper, we present some practical necessary and sufficient conditions for the existence of an \(\eta \)-Hermitian solution to a system of constrained two-sided coupled real quaternion matrix equations and provide the general \(\eta \)-Hermitian solution to the system when it is solvable. Moreover, we present an algorithm and a numerical example to illustrate our results.
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Consistency of Quaternion Matrix Equations $AX^{\star}-XB=C$ and $X-AX^\star B=C$
The Electronic Journal of Linear Algebra, 2019Co-Authors: Xin Liu, Qing-wen Wang, Yang ZhangAbstract:For a given ordered units triple $\{q_1, q_2, q_3\}$, the solutions to the quaternion matrix equations $AX^{\star}-XB=C$ and $X-AX^{\star}B=C$, $X^{\star} \in \{ X , X^{\eta} , X^* , X^{\eta*}\}$, where $X^*$ is the Conjugate Transpose of $X$, $X^{\eta}=-\eta X \eta$ and $X^{\eta*}=-\eta X^* \eta$, $\eta \in \{q_1, q_2, q_3\}$, are discussed. Some new real representations of quaternion matrices are used, which enable one to convert $\eta$-Conjugate (Transpose) matrix equations into some real matrix equations. By using this idea, conditions for the existence and uniqueness of solutions to the above quaternion matrix equations are derived. Also, methods to construct the solutions from some related real matrix equations are presented.
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The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation
Mathematical Problems in Engineering, 2017Co-Authors: Xin Liu, Qing-wen WangAbstract:For a given generalized reflection matrix , that is, , , where is the Conjugate Transpose matrix of , a matrix is called a Hermitian (anti)reflexive matrix with respect to if and By using the Kronecker product, we derive the explicit expression of least squares Hermitian (anti)reflexive solution with the least norm to matrix equation over complex field.
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The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation AXB=C
Hindawi Limited, 2017Co-Authors: Xin Liu, Qing-wen WangAbstract:For a given generalized reflection matrix J, that is, JH=J, J2=I, where JH is the Conjugate Transpose matrix of J, a matrix A∈Cn×n is called a Hermitian (anti)reflexive matrix with respect to J if AH=A and A=±JAJ. By using the Kronecker product, we derive the explicit expression of least squares Hermitian (anti)reflexive solution with the least norm to matrix equation AXB=C over complex field
Mingsong Cheng - One of the best experts on this subject based on the ideXlab platform.
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Perturbation Analysis of the Mixed-Type Lyapunov Equation
Lecture Notes in Electrical Engineering, 2011Co-Authors: Mingsong ChengAbstract:This paper concerns the mixed-type Lyapunov equation \(X=A^*XB+B^*XA+Q,\) where \(A,B,\) and \(Q\) are \(n\times n\) complex matrices and \(A^*\) the Conjugate Transpose of a matrix \(A.\) A perturbation bound for the solution to this matrix equation is derived, an explicit expression of the condition number is obtained, and the backward error of an approximate solution is evaluated by using the techniques developed in Sun (Linear Algebra Appl 259:183–208, 1997), Sun and Xu (Linear Algebra Appl 362:211–228, 2003). The results are illustrated by using some numerical examples.
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PERTURBATION ANALYSIS OF A NONLINEAR MATRIX EQUATION
Taiwanese Journal of Mathematics, 2006Co-Authors: Mingsong ChengAbstract:Consider the nonlinear matrix equation $X+A^*X^{-2}A=I$, where $A$ is an $n\times n$ complex matrix, $I$ the identity matrix and $A^*$ the Conjugate Transpose of the matrix $A$. In this paper a perturbation bound for a class of special solutions of this matrix equation is derived, and an explicit expression of its condition number is obtained. The results are illustrated by using some numerical examples.
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perturbation analysis of the hermitian positive definite solution of the matrix equation x a x 2a i
Linear Algebra and its Applications, 2005Co-Authors: Mingsong Cheng, Shufang XuAbstract:Abstract Consider the nonlinear matrix equation X − A*X−2A = I, where A is an n × n complex matrix, I the identity matrix and A* the Conjugate Transpose of a matrix A. In this paper, it is proved that this matrix equation has a unique Hermitian positive definite solution provided ∥A∥2
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Perturbation analysis of the Hermitian positive definite solution of the matrix equation X − A*X−2A = I
Linear Algebra and its Applications, 2005Co-Authors: Mingsong ChengAbstract:Abstract Consider the nonlinear matrix equation X − A*X−2A = I, where A is an n × n complex matrix, I the identity matrix and A* the Conjugate Transpose of a matrix A. In this paper, it is proved that this matrix equation has a unique Hermitian positive definite solution provided ∥A∥2
Fernando De Terán - One of the best experts on this subject based on the ideXlab platform.
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Uniqueness of solution of a generalized ⋆-Sylvester matrix equation
Linear Algebra and its Applications, 2016Co-Authors: Fernando De Terán, Bruno IannazzoAbstract:Abstract We present necessary and sufficient conditions for the existence of a unique solution of the generalized ⋆-Sylvester matrix equation A X B + C X ⋆ D = E , where A , B , C , D , E are square matrices of the same size with real or complex entries, and where ⋆ stands for either the Transpose or the Conjugate Transpose. This generalizes several previous uniqueness results for specific equations like the ⋆-Sylvester or the ⋆-Stein equations.
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A note on the consistency of a system of $\star$-Sylvester equations
arXiv: Numerical Analysis, 2014Co-Authors: Fernando De TeránAbstract:Let ${\mathbb F}$ be a field with characteristic not $2$, and $A_i\in{\mathbb F}^{m\times n},B_i\in{\mathbb F}^{n\times m},C_i\in{\mathbb F}^{m\times m}$, for $i=1,...,k$. In this short note, we obtain necessary and sufficient conditions for the consistency of the system of $\star$-Sylvester equations $A_iX-X^\star B_i=C_i$, for $i=1,...,k$, where $\star$ denotes either the Transpose or, when ${\mathbb F}={\mathbb C}$, the Conjugate Transpose.
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The solution of the equation AX + BX ⋆ = 0
Linear and Multilinear Algebra, 2013Co-Authors: Fernando De TeránAbstract:We give a complete solution of the matrix equation AX + BX ⋆ = 0, where A, B ∈ ℂ m×n are two given matrices, X ∈ ℂ n×n is an unknown matrix, and ⋆ denotes the Transpose or the Conjugate Transpose. We provide a closed formula for the dimension of the solution space of the equation in terms of the Kronecker canonical form of the matrix pencil A + λB, and we also provide an expression for the solution X in terms of this canonical form, together with two invertible matrices leading A + λB to the canonical form by strict equivalence.
Jigunag Sun - One of the best experts on this subject based on the ideXlab platform.
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Perturbation analysis of the maximal solution of the matrix equation X+A∗X−1A=P. II
Linear Algebra and its Applications, 2003Co-Authors: Jigunag SunAbstract:AbstractConsider the nonlinear matrix equationX+A∗X−1A=P,where A,P are n×n complex matrices with P Hermitian positive definite, and A* denotes the Conjugate Transpose of a matrix A. In this paper a sharper perturbation bound for the maximal solution to the matrix equation is derived, explicit expressions of the condition number for the maximal solution are obtained, and the backward error of an approximate solution to the maximal solution is evaluated by using the techniques developed in [Linear Algebra Appl. 259 (1997) 183; Linear Algebra Appl. 350 (2002) 237]. The results are illustrated by using numerical examples
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perturbation analysis of the maximal solution of the matrix equation x a x 1a p
Linear Algebra and its Applications, 2001Co-Authors: Jigunag SunAbstract:Consider the nonlinear matrix equation X + A*X(-1)A = P, where A, P are n x n complex matrices with P Hermitian positive definite, and A* denotes the Conjugate Transpose of a matrix A. In this pape ...
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Perturbation analysis of the maximal solution of the matrix equation X+A∗X−1A=P
Linear Algebra and its Applications, 2001Co-Authors: Jigunag SunAbstract:Consider the nonlinear matrix equation X + A*X(-1)A = P, where A, P are n x n complex matrices with P Hermitian positive definite, and A* denotes the Conjugate Transpose of a matrix A. In this pape ...
