共查询到19条相似文献,搜索用时 140 毫秒
1.
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
2.
Zhen-yunPeng Xi-yanHu LeiZhang 《计算数学(英文版)》2004,22(6):873-880
The necessary and sufficient conditions for the existence of and the expressions for the bisymmetric solutions of the matrix equations (Ⅰ)A1X1B1 A2X2B2 ^… AkXkBk=D,(Ⅱ)A1XB1 A2XB2 … AkXBk=D and (Ⅲ) (A1XB1,A2XB2,…,AkXBk)=(D1,D2,…,Dk) are derived by using Kronecker product and Moore-Penrose generalized inverse of matrices. In addition, in corresponding solution set of the matrix equations, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm is given. Numerical methods and numerical experiments of finding the neaxest solutions axe also provided. 相似文献
3.
体上一矩阵方程组的再研究 总被引:1,自引:1,他引:0
Let F,Qand Q be an arbitrary skew field,a skew field with an involutorial antiautomorphism,a strong p-division ring,respectively.This paper is sequel to [1].A practical method of finding the general solution of the system of matrix equations over F is presented.The necessary and sufficient conditions for the existence of and the explicit expressions for the self-conjugate matrix solution over Q,the positive semidefinite self-conjugate matrix solution over Q of the system are derived. 相似文献
4.
In this paper, nonlinear matrix equations of the form X + A*f1 (X)A + B*f2 (X)B = Q are discussed. Some necessary and sufficient conditions for the existence of solutions for this equation are derived. It is shown that under some conditions this equation has a unique solution, and an iterative method is proposed to obtain this unique solution. Finally, a numerical example is given to identify the efficiency of the results obtained. 相似文献
5.
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. 相似文献
6.
In this article, the generalized reflexive solution of matrix equations (AX = B, XC = D) is considered. With special properties of generalized reflexive matrices, the necessary and sufficient conditions for the solvability and the general expression of the solution are obtained. Moreover, the related optimal approximation problem to a given matrix over the solution set is solved. 相似文献
7.
关于任意体上的一个矩阵方程组 总被引:2,自引:0,他引:2
In this peper,we give necessary and sufficient conditions for solvabilty of the system of matrix equations over an arbitrary skew field .The distinct expressions of the general solutions of the systerm are found and some properties of the soultons are also given.Thereupon the original projection matrix is generalized over an arbitrary skew field. 相似文献
8.
This paper discusses the solutions of the linear matrix equation B~T XB=D on some linear manifolds. Some necessary and sufficient conditions for the existence of the solution and the expression of the general solution are given. And also some optimal approximation solutions are discussed. 相似文献
9.
Cheng-yi Zhang Fengmin Xu Zongben Xu Jicheng Li 《Frontiers of Mathematics in China》2014,9(5):1141-1168
The definitions of θ-ray pattern proposed to establish some new results matrix and θ-ray matrix are firstly on nonsingularity/singularity and convergence of general H-matrices. Then some conditions on the matrix A ∈ C^n×n and nonempty α (n) = {1,2,... ,n} are proposed such that A is an invertible H-matrix if A(α) and A/α are both invertible H-matrices. Furthermore, the important results on Schur complement for general H-matrices are presented to give the different necessary and sufficient conditions for the matrix A E HM and the subset α C (n) such that the Schur complement matrix A/α∈ HI^n-|α| or A/α ∈ Hn-|α|^M or A/α ∈ H^n-| α|^S. 相似文献
10.
In this paper, the Hermitian positive definite solutions of the nonlinear matrix equation X^s - A^*X^-tA = Q are studied, where Q is a Hermitian positive definite matrix, s and t are positive integers. The existence of a Hermitian positive definite solution is proved. A sufficient condition for the equation to have a unique Hermitian positive definite solution is given. Some estimates of the Hermitian positive definite solutions are obtained. Moreover, two perturbation bounds for the Hermitian positive definite solutions are derived and the results are illustrated by some numerical examples. 相似文献
11.
该文给出了四元数矩阵方程组X_1B_1=C_1,X_2B_2=C2,A_1X_1B_3+A_2X_2B_4=C_b可解的充要条件及其通解的表达式,利用此结果建立了四元数矩阵方程组XB_a=C_a,A_bXB_b=C_b有广义(反)反射解的充要条件及其有此种解时通解的表达式. 相似文献
12.
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
13.
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
14.
Jian-hui Long Xi-yan Hu Lei Zhang 《Bulletin of the Brazilian Mathematical Society》2008,39(3):371-386
In this paper, we study the matrix equation X + A*X
−1
A + B*X
−1
B = I, where A, B are square matrices, and obtain some conditions for the existence of the positive definite solution of this equation. Two
iterative algorithms to find the positive definite solution are given. Some numerical results are reported to illustrate the
effectiveness of the algorithms.
This research supported by the National Natural Science Foundation of China 10571047 and Doctorate Foundation of the Ministry
of Education of China 20060532014. 相似文献
15.
Huang Liping 《数学学报(英文版)》1998,14(1):91-98
LetH
F
be the generalized quaternion division algebra over a fieldF with charF#2. In this paper, the adjoint matrix of anyn×n matrix overH
F
[γ] is defined and its properties is discussed. By using the adjoint matrix and the method of representation matrix, this
paper obtains several necessary and sufficient conditions for the existence of a solution or a unique solution to the matrix
equation Σ
i=0
k
A
i
XB
i
=E overH
F
, and gives some explicit formulas of solutions.
Supported by the National Natural Science Foundation of China and Human 相似文献
16.
Yuan‐Bei Deng Zhong‐Zhi Bai Yong‐Hua Gao 《Numerical Linear Algebra with Applications》2006,13(10):801-823
The consistent conditions and the general expressions about the Hermitian solutions of the linear matrix equations AXB=C and (AX, XB)=(C, D) are studied in depth, where A, B, C and D are given matrices of suitable sizes. The Hermitian minimum F‐norm solutions are obtained for the matrix equations AXB=C and (AX, XB)=(C, D) by Moore–Penrose generalized inverse, respectively. For both matrix equations, we design iterative methods according to the fundamental idea of the classical conjugate direction method for the standard system of linear equations. Numerical results show that these iterative methods are feasible and effective in actual computations of the solutions of the above‐mentioned two matrix equations. Copyright © 2006 John Wiley & Sons, Ltd. 相似文献
17.
黄礼平 《高校应用数学学报(英文版)》2002,17(1):109-118
By using the Smith normal form of polynomial matrix and algebraic methods, this paper discusses the solvability for the linear matrix equation ∑AiXBi = C over a field, and obtains the explicit formulas of general solution or unique solution. 相似文献
18.
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
19.
S. A. Amitsur 《Israel Journal of Mathematics》1974,17(3):241-247
LetA=k (X
1, X2..., Xm) be the division ring generated by genericn×n matrices over a fieldk; thenA is not a crossed product in the following cases: (i) there exists a primeq such thatq
3ℛn;(ii)[k:Q]=m, whereQ is the field of rationals, then if eitherq
3ℛn for someq for whichq-1ℛm, orq
2/nn for some other prime; (iii)k=Z
p
r a finite field ofp
r elements and eitherq
3ℛn for sameqℛp
r-1 orq
2ℛn for some other primes. Other cases are also considered. 相似文献