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提出了四元数矩阵的一种实向量表示法,可以结合矩阵的半张量积研究四元数矩阵方程.给出了四元数矩阵方程X-AXB=CY+D的最小二乘Hermitian解的通解表达式,以及该方程具有Hermitian解的充要条件,通过数值实验,验证该方法的有效性. 相似文献
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田永革 《应用数学与计算数学学报》1992,6(1):46-52
本文利用矩阵的广义逆给出了任意域,上齐次线性矩阵方程组A_1X_1B_1=A_2X_2B_2=…=A_kX_kB_k解的通式,并在此基础之上讨论了一类矩阵集合 相似文献
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确立了一类分块矩阵M11 M12 XM21 M22 M23Y M32 M33的最大秩公式,其中,X和Y是两个受限于四元数线性矩阵方程A_1X=C_1,XB_1=C_2,A_3XB_3=C_3,A_2Y=D_1,YB_2=D_2.的变量矩阵。作为该公式的一项应用,我们推导出上述矩阵方程解集等同于另一四元数二次矩阵方程组解集的条件。 相似文献
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李璟 《应用数学与计算数学学报》2014,(4):493-501
研究了包含η-厄尔米特矩阵的四元数矩阵方程组.用四元数矩阵的秩和广义逆给出了一个包含η-厄尔米特矩阵的四元数矩阵方程组相容的充分必要条件.进一步地,用四元数矩阵的广义逆给出了这个四元数矩阵方程组的通解表达式. 相似文献
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《数学的实践与认识》2017,(24)
把实数域上的辛矩阵概念推广到四元数体上形成共轭辛矩阵类.用矩阵四分块形式刻划了正定辛矩阵和自共轭辛矩阵的特征结构.作为应用,给出四元数矩阵方程AS=B存在四分块对角型共轭辛矩阵解的充要条件及其解的表达式,同时用数值算例说明所给方法的可行性. 相似文献
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李怡君王卿文 《应用数学与计算数学学报》2018,(3):598-607
基于广义Sylvester实圆元数矩阵方程组的解■当A_i,B_i和C_i(i=1,2,3)是被复数矩阵给定的,X,Y,Z和W是可变矩阵.计算耦合广义S_ylvester实四元数矩阵方程组的通解W的秩的极值. 相似文献
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借助于四元数体上自共轭矩阵的奇异值分解,给出了四元数矩阵方程AX+XB+CXD=F的极小范数最小二乘解.同时,在有解的条件下给出了Hermite最小二乘解及其通解的表达形式. 相似文献
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In this article we establish necessary and sufficient conditions for the existence and the expressions of the general real solutions to the classical system of quaternion matrix equations A 1 XB 1 = C 1, A 2 XB 2 = C 2. Moreover, formulas of the maximal and minimal ranks of four real matrices X 1, X 2, X 3, and X 4 in solution X = X 1 + X 2 i + X 3 j + X 4 k to the system mentioned above are derived. As applications, we give necessary and sufficient conditions for the quaternion matrix equations A 1 XB 1 = C 1, A 2 XB 2 = C 2, A 3 XB 3 = C 3 to have common real solutions. In addition, the maximal and minimal ranks of four real matrices E, F, G, and H in the common generalized inverse of A 1 + B 1 i + C 1 j + D 1 k and A 2 + B 2 i + C 2 j + D 2 k, which can be expressed as E + Fi + Gj + Hk are also presented. 相似文献
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Yongge Tian 《Linear and Multilinear Algebra》2013,61(2):123-147
The solvability conditions of the following two linear matrix equations (i)A1X1B1 +A2X2B2 +A3X3B3 =C,(ii) A1XB1 =C1 A2XB2 =C2 are established using ranks and generalized inverses of matrices. In addition, the duality of the three types of matrix equations (iii) A 1 X 1 B 1+A 2 X 2 B 2+A 3 X 3 B 3+A 4 X 4 B 4=C, (iv) A 1 XB 1=C 1 A 2 XB 2=C 2 A 3 XB 3=C 3 A 4 XB 4=C 4, (v) AXB+CXD=E are also considered. 相似文献
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In this paper, we study the solvability of the operator equations A*X + X*A = C and A*XB + B*X*A = C for general adjointable operators on Hilbert C*-modules whose ranges may not be closed. Based on these results we discuss the solution to the operator equation AXB = C, and obtain some necessary and sufficient conditions for the existence of a real positive solution, of a solution X with B*(X* + X)B ≥ 0, and of a solution X with B*XB ≥ 0. Furthermore in the special case that R(B) í [`(R(A*))]{R(B)\subseteq\overline{R(A*)}} we obtain a necessary and sufficient condition for the existence of a positive solution to the equation AXB = C. The above results generalize some recent results concerning the equations for operators with closed ranges. 相似文献
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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. 相似文献
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Somayeh Rashedi Ghodrat Ebadi Anjan Biswas 《Journal of the Egyptian Mathematical Society》2013,21(3):175-183
In this paper, we establish the formulas of the extermal ranks of the quaternion matrix expression f(X1, X2) = C7 ? A4X1B4 ? A5X2B5 where X1, X2 are variant quaternion matrices subject to quaternion matrix equations A1X1 = C1, A2X1 = C2, A3X1 = C3, X2B1 = C4, X2B2 = C5, X2B3 = C6. As applications, we give a new necessary and sufficient condition for the existence of solutions to some systems of quaternion matrix equations. Some results can be viewed as special cases of the results of this paper. 相似文献
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In this paper, we discuss the generalized quaternion matrix equation AXB+CX⋆D=E, where X⋆ is one of X, X*, the η-conjugate or the η-conjugate transpose of X with η∈{i,j,k}. Two new real representations of a generalized quaternion matrix are proposed. By using this method, the criteria for the existence and uniqueness of solutions to the mentioned matrix equation as well as the existence of X=± X⋆ solutions to the generalized quaternion matrix equation AXB+CXD=E are derived in a unified way. 相似文献
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A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications 总被引:3,自引:0,他引:3
QingWenWANG 《数学学报(英文版)》2005,21(2):323-334
We consider the system of four linear matrix equations A1X = C1, XB2=C2, A3XB3=C3 and A4XB4 = C4 over h, an arbitrary von Neumann regular ring with identity. A necessary and sufficient condition for the existence and the expression of the general solution to the system are derived. As applications, necessary and sufficient conditions are given for the system of matrix equations A1X = C1 and A3X=C3 to have a bisymmetric solution, the system of matrix equations A1X = C1 and A3XB3 = C3 to have a perselfconjugate solution over h with an involution and char h≠2, respectively. The representations of such solutions are also presented. Moreover, some auxiliary resultson other systems over h are obtained. The previous known results on some systems of matrix equations are special cases of the new results. 相似文献
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Zhiping XiongYingying Qin 《Applied mathematics and computation》2011,218(7):3330-3337
In this article, we consider common Re-nnd and Re-pd solutions of the matrix equations AX = C and XB = D with respect to X, where A, B, C and D are given matrices. We give necessary and sufficient conditions for the existence of common Re-nnd and Re-pd solutions to the pair of the matrix equations and derive a representation of the common Re-nnd and Re-pd solutions to these two equations when they exist. The presented examples show the advantage of the proposed approach. 相似文献
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On derivable mappings 总被引:1,自引:0,他引:1
Jiankui Li 《Journal of Mathematical Analysis and Applications》2011,374(1):311-322
A linear mapping δ from an algebra A into an A-bimodule M is called derivable at c∈A if δ(a)b+aδ(b)=δ(c) for all a,b∈A with ab=c. For a norm-closed unital subalgebra A of operators on a Banach space X, we show that if C∈A has a right inverse in B(X) and the linear span of the range of rank-one operators in A is dense in X then the only derivable mappings at C from A into B(X) are derivations; in particular the result holds for all completely distributive subspace lattice algebras, J-subspace lattice algebras, and norm-closed unital standard algebras of B(X). As an application, every Jordan derivation from such an algebra into B(X) is a derivation. For a large class of reflexive algebras A on a Banach space X, we show that inner derivations from A into B(X) can be characterized by boundedness and derivability at any fixed C∈A, provided C has a right inverse in B(X). We also show that if A is a canonical subalgebra of an AF C∗-algebra B and M is a unital Banach A-bimodule, then every bounded local derivation from A into M is a derivation; moreover, every bounded linear mapping from A into B that is derivable at the unit I is a derivation. 相似文献
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Singular values, norms, and commutators 总被引:1,自引:0,他引:1
Omar Hirzallah 《Linear algebra and its applications》2010,432(5):1322-1336
Let and Xi, i=1,…,n, be bounded linear operators on a separable Hilbert space such that Xi is compact for i=1,…,n. It is shown that the singular values of are dominated by those of , where ‖·‖ is the usual operator norm. Among other applications of this inequality, we prove that if A and B are self-adjoint operators such that a1?A?a2 and b1?B?b2 for some real numbers and b2, and if X is compact, then the singular values of the generalized commutator AX-XB are dominated by those of max(b2-a1,a2-b1)(X⊕X). This inequality proves a recent conjecture concerning the singular values of commutators. Several inequalities for norms of commutators are also given. 相似文献