• 论文 •

广义对称约束条件下矩阵表达式A-BXC 的极秩问题

1. 天水师范学院数学与统计学院, 甘肃 天水 741001
• 收稿日期:2014-05-16 修回日期:2014-10-15 出版日期:2015-02-20 发布日期:2015-01-27
• 通讯作者: 梁茂林(1981-),男,硕士,讲师,研究方向为数值线性代数. E-mail:liangml2005@163.com E-mail:liangml2005@163.com
• 作者简介:代丽芳(1981-),女,硕士,讲师,研究方向为非线性泛函分析. E-mail:dailf2005@163.com
• 基金资助:
甘肃省教育厅项目(1108B-03);天水师范学院青年基金项目(TSA1315)

The minimal and maximal ranks problems of matrix expression A-BXC under generalized symmetric constraints

DAI Li-fang, LIANG Mao-lin, HE Wan-sheng

1. School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, Gansu, China
• Received:2014-05-16 Revised:2014-10-15 Online:2015-02-20 Published:2015-01-27

Abstract: Let R,S be generalized reflection matrices, that is, R*=R, R2=I, S*=S, S2=I. A matrix X is called generalized reflective matrix (generalized skew-reflective matrix) if RXS=X(RXS=-X). The minimal and maximal ranks problems of matrix expression A-BXC with generalized reflective or skew-reflective matrices X are studied, and some solvability conditions of matrix equation BXC=A are derived.

• O241.6
 [1] CHEN H C. Generalized reflexive matrices: special properties and applications[J]. SIAM J Matrix Anal Appl, 1998, 19(1):140-153. [2] 陈新. 机械结构动态设计理论方法及应用[M]. 北京:机械工业出版社,1997. CHEN Xin. The dynamic design of mechanical structure: theory, method and applications[M]. Beijing: Machinery Industry Press, 1997. [3] DAI Hua. On the symmetric solutions of linear matrix equations[J]. Linear Algebra Appl, 1990, 131:1-7. [4] CHU K E. Symmetric solutions of linear matrix equations by matrix decompositions[J]. Linear Algebra Appl, 1989, 119:35-50. [5] GROB J. Nonnegative-definite and positive-definite solutions to the matrix equation AXB=C-revisited[J]. Linear Algebra Appl, 2000, 321:123-129. [6] CHANG Xiaowen, WANG J. The symmetric solution of the matrix equations AY+YA=C, AXAT+BYBT=C and (ATXA, BTXB)=(C,D)[J]. Linear Algebra Appl, 1993, 179:171-189. [7] XU Guiping, WEI Musheng, ZHENG Daosheng. On solution of matrix equation AXB+CYD=F[J]. Linear Algebra Appl, 1998, 279:93-109. [8] WANG Guorong, WEI Yimin, QIAO Sanzheng. Generalized inverse: theory and computations[M]. Beijing: Science Press, 2004. [9] 彭亚新. 求解约束矩阵方程及其最佳逼近的迭代法的研究[D].长沙:湖南大学,2004. PENG Yaxin. The research on the iterative methods for solving constrained matrix equations and the associated optimal approximation[D]. Changsha: Hunan University, 2004. [10] DENG Yuanbei, BAI Zhongzhi, GAO Yonghua. Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations[J]. Numer. Linear Algebra Appl, 2006, 13:801-823. [11] CHU K E. Singular value and generalized singular value decompositions and the solution of linear matrix equations[J]. Linear Algebra Appl, 1987, 88/89:83-98. [12] TIAN Yongge. The maximal and minimal ranks of some expressions of generalized inverses of matrices[J]. Southeast Asian Bull Math, 2002, 25:745-755. [13] TIAN Yongge. Upper and lower bounds for ranks of matrix expressions using generalized inverses[J]. Linear Algebra Appl, 2002, 355:187-214. [14] TIAN Yongge. Some properties of submatrices in a solution to the matrix equation AXB=C with applications[J]. J Franklin Inst, 2009, 346:557-569. [15] LIANG Maolin, DAI Lifang. The left and right inverse eigenvalue problems of generalized reflexive and anti-reflexive matrices[J]. J Comput Appl Math, 2010, 234:743-749. [16] TIAN Yongge. The maximal and minimal ranks of A-BXC with applications[J]. New York J Math, 2003, 9:345-362. [17] GUO Wenbin, HUANG Tingzhu. Extremal ranks of matrix expression of A-BXC with respect to Hermitian matrix[J]. Appl Math Comput, 2010, 217:2381-2389. [18] WANG Hongxing. The minimal rank of A-BX with respect to Hermitian matrix[J]. Appl Math Comput, 2014, 233:55-61.
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