Generally unitary solution to the system of martix equations over the quaternion field [X mA ns =B ns ,X nn C nt =D nt ] is considered. A necessary and sufficient condition for the existence o...Generally unitary solution to the system of martix equations over the quaternion field [X mA ns =B ns ,X nn C nt =D nt ] is considered. A necessary and sufficient condition for the existence of and the expression for the generally unitary solution of the system are derived.展开更多
Minor self conjugate (msc) and skewpositive semidefinite (ssd) solutions to the system of matrix equations over skew fields [A mn X nn =A mn ,B sn X nn =O sn ] are considered. Necessary and su...Minor self conjugate (msc) and skewpositive semidefinite (ssd) solutions to the system of matrix equations over skew fields [A mn X nn =A mn ,B sn X nn =O sn ] are considered. Necessary and sufficient conditions for the existence of and the expressions for the msc solutions and the ssd solutions are obtained for the system.展开更多
We derive necessary and sufficient conditions for the existence and an expression of the (anti)reflexive solution with respect to the nontrivial generalized reflection matrix P to the system of complex matrix equati...We derive necessary and sufficient conditions for the existence and an expression of the (anti)reflexive solution with respect to the nontrivial generalized reflection matrix P to the system of complex matrix equations AX = B and XC = D. The explicit solutions of the approximation problem min x∈Ф ||X - E||F was given, where E is a given complex matrix and Ф is the set of all reflexive (or antireflexive) solutions of the system mentioned above, and ||·|| is the Frobenius norm. Furthermore, it was pointed that some results in a recent paper are special cases of this paper.展开更多
Necessary and sufficient conditions are given for the existence of the general solution, the centrosymmetric solution, and the centroskewsymmetric solution to a system of linear matrix equations over an arbitrary skew...Necessary and sufficient conditions are given for the existence of the general solution, the centrosymmetric solution, and the centroskewsymmetric solution to a system of linear matrix equations over an arbitrary skew field. The representations of such the solutions of the system are also derived.展开更多
In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new re...In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new result.展开更多
Let P∈C^(n×n)be a Hermitian and{k+1}-potent matrix,i.e.,P^(k+1)=P=P^(*),where(·)^(*)stands for the conjugate transpose of a matrix.A matrix X∈C^(n×n)is called{P,k+1}-reflexive(anti-reflexive)if PXP=X(...Let P∈C^(n×n)be a Hermitian and{k+1}-potent matrix,i.e.,P^(k+1)=P=P^(*),where(·)^(*)stands for the conjugate transpose of a matrix.A matrix X∈C^(n×n)is called{P,k+1}-reflexive(anti-reflexive)if PXP=X(P XP=-X).The system of matrix equations AX=C,XB=D subject to{P,k+1}-reflexive and anti-reflexive constraints are studied by converting into two simpler cases:k=1 and k=2,the least squares solution and the associated optimal approximation problem are also considered.展开更多
We consider the system of four linear matrix equations A_1 X = C_1,XB_2 =C_2,A_3,XB_3, = C3 and A_4XB_4 = C_4 over R, an arbitrary von Neumann regular ring with identity. Anecessary and sufficient condition for the ex...We consider the system of four linear matrix equations A_1 X = C_1,XB_2 =C_2,A_3,XB_3, = C3 and A_4XB_4 = C_4 over R, an arbitrary von Neumann regular ring with identity. Anecessary and sufficient condition for the existence and the expression of the general solution tothe system are derived. As applications, necessary and sufficient conditions are given for thesystem of matrix equations A_1X = C_1 and A_3X = C_3 to have a bisymmetric solution, the system ofmatrix equations A_1X = C_1 and A_3XB_3 = C_3 to have a perselfconjugate solution over R with aninvolution and char R≠2, respectively. The representations of such solutions are also presented.Moreover, some auxiliary results on other systems over R are obtained. The previous known results onsome systems of matrix equations are special cases of the new results.展开更多
A new expression is established for the common solution to six classical linear quaternion matrix equations A 1 X = C 1 , X B 1 = C 3 , A 2 X = C 2 , X B 2 = C 4 , A 3 X B 3 = C 5 , A 4 X B 4 = C 6 which was investiga...A new expression is established for the common solution to six classical linear quaternion matrix equations A 1 X = C 1 , X B 1 = C 3 , A 2 X = C 2 , X B 2 = C 4 , A 3 X B 3 = C 5 , A 4 X B 4 = C 6 which was investigated recently by Wang, Chang and Ning (Q. Wang, H. Chang, Q. Ning, The common solution to six quaternion matrix equations with applications, Appl. Math. Comput. 195: 721-732 (2008)). Formulas are derived for the maximal and minimal ranks of the common solution to this system. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition is presented for the invariance of the rank of the general solution to this system. Some known results can be regarded as the special cases of the results in this paper.展开更多
Let Ω be a finite dimensional central algebra with an involutorial anti-automorphism and chartΩ≠2.Two systems of matrix equations over Ω are consid-ered.Necessary and sufficient conditions for the existences of ge...Let Ω be a finite dimensional central algebra with an involutorial anti-automorphism and chartΩ≠2.Two systems of matrix equations over Ω are consid-ered.Necessary and sufficient conditions for the existences of general solutions,andper(skew)selfconjugate solutions of the systems are given,respectively.展开更多
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 ...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.展开更多
文摘Generally unitary solution to the system of martix equations over the quaternion field [X mA ns =B ns ,X nn C nt =D nt ] is considered. A necessary and sufficient condition for the existence of and the expression for the generally unitary solution of the system are derived.
文摘Minor self conjugate (msc) and skewpositive semidefinite (ssd) solutions to the system of matrix equations over skew fields [A mn X nn =A mn ,B sn X nn =O sn ] are considered. Necessary and sufficient conditions for the existence of and the expressions for the msc solutions and the ssd solutions are obtained for the system.
基金supported by the National Natural Science Foundation of China (Grant No.60672160)
文摘We derive necessary and sufficient conditions for the existence and an expression of the (anti)reflexive solution with respect to the nontrivial generalized reflection matrix P to the system of complex matrix equations AX = B and XC = D. The explicit solutions of the approximation problem min x∈Ф ||X - E||F was given, where E is a given complex matrix and Ф is the set of all reflexive (or antireflexive) solutions of the system mentioned above, and ||·|| is the Frobenius norm. Furthermore, it was pointed that some results in a recent paper are special cases of this paper.
基金Supported by the National Natural Science Foundation of China(10471085)
文摘Necessary and sufficient conditions are given for the existence of the general solution, the centrosymmetric solution, and the centroskewsymmetric solution to a system of linear matrix equations over an arbitrary skew field. The representations of such the solutions of the system are also derived.
基金Project supported by the National Natural Science Foundation of China (Grant No.60672160)
文摘In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new result.
基金Supported by the Education Department Foundation of Hebei Province(QN2015218)Supported by the Natural Science Foundation of Hebei Province(A2015403050)
文摘Let P∈C^(n×n)be a Hermitian and{k+1}-potent matrix,i.e.,P^(k+1)=P=P^(*),where(·)^(*)stands for the conjugate transpose of a matrix.A matrix X∈C^(n×n)is called{P,k+1}-reflexive(anti-reflexive)if PXP=X(P XP=-X).The system of matrix equations AX=C,XB=D subject to{P,k+1}-reflexive and anti-reflexive constraints are studied by converting into two simpler cases:k=1 and k=2,the least squares solution and the associated optimal approximation problem are also considered.
基金This research is supported by the Natural Science Foundation of China(No.0471085the Natural Science Foundation of Shanghai)the Development Foundation of Shanghai Educational Committee the Special Funds for Major Specialities of Shanghai Education Co
文摘We consider the system of four linear matrix equations A_1 X = C_1,XB_2 =C_2,A_3,XB_3, = C3 and A_4XB_4 = C_4 over R, an arbitrary von Neumann regular ring with identity. Anecessary and sufficient condition for the existence and the expression of the general solution tothe system are derived. As applications, necessary and sufficient conditions are given for thesystem of matrix equations A_1X = C_1 and A_3X = C_3 to have a bisymmetric solution, the system ofmatrix equations A_1X = C_1 and A_3XB_3 = C_3 to have a perselfconjugate solution over R with aninvolution and char R≠2, respectively. The representations of such solutions are also presented.Moreover, some auxiliary results on other systems over R are obtained. The previous known results onsome systems of matrix equations are special cases of the new results.
基金Supported by the National Natural Science Foundation of Shanghai (No. 11ZR1412500)the Ph.D. Programs Foundation of Ministry of Education of China (No. 20093108110001)Shanghai Leading Academic Discipline Project (No. J50101)
文摘A new expression is established for the common solution to six classical linear quaternion matrix equations A 1 X = C 1 , X B 1 = C 3 , A 2 X = C 2 , X B 2 = C 4 , A 3 X B 3 = C 5 , A 4 X B 4 = C 6 which was investigated recently by Wang, Chang and Ning (Q. Wang, H. Chang, Q. Ning, The common solution to six quaternion matrix equations with applications, Appl. Math. Comput. 195: 721-732 (2008)). Formulas are derived for the maximal and minimal ranks of the common solution to this system. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition is presented for the invariance of the rank of the general solution to this system. Some known results can be regarded as the special cases of the results in this paper.
基金Foundation item:Supported by the National Natural Science Foundation of China(10071078)Natural Science Foundation of Shandong Province(Q99A08)
文摘Let Ω be a finite dimensional central algebra with an involutorial anti-automorphism and chartΩ≠2.Two systems of matrix equations over Ω are consid-ered.Necessary and sufficient conditions for the existences of general solutions,andper(skew)selfconjugate solutions of the systems are given,respectively.
基金This research was supported by the grants from the National Natural Science Foundation of China (11571220, 11171205).
文摘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.