$\mathfrak{m}$-WG 逆的性质和计算

数学物理学报 ›› 2024, Vol. 44 ›› Issue (3): 547-562.

$\mathfrak{m}$-WG 逆的性质和计算

韦华全¹(),吴辉^1,^*(),刘晓冀²(),靳宏伟³()

1.广西大学数学与信息科学学院南宁 530004
2.广西职业师范学院教育学院南宁 530007
3.广西民族大学数学与物理学院南宁 530006

收稿日期:2023-02-06 修回日期:2023-10-06 出版日期:2024-06-26 发布日期:2024-05-17
通讯作者: *吴辉, Email：huiwumath168@163.com
作者简介:韦华全, Email：weihuaquan@163.com;|刘晓冀, Email：xiaojiliu72@126.com;|靳宏伟, Email：hw-jin@hotmail.com
基金资助:
国家自然科学基金(12061011);广西自然科学基金(2020GXNSFDA238014);广西自然科学基金(2023JJA110104);广西高校中青年教师科研基础能力提升项目(2022KY0610);广西研究生教育创新计划资助项目(YCBZ2023021)

Properties and Computations of the $\mathfrak{m}$-WG Inverse

Wei Huaquan¹(),Wu Hui^1,^*(),Liu Xiaoji²(),Jin Hongwei³()

1. School of Mathematics and Information Science, Guangxi University, Nanning 530004
2. School of Education, Guangxi Vocational Normal University, Nanning 530007
3. School of Mathematics and Physics, Guangxi Minzu University, Nanning 530006

Received:2023-02-06 Revised:2023-10-06 Online:2024-06-26 Published:2024-05-17
Supported by:
NSF of China(12061011);NSF of Guangxi(2020GXNSFDA238014);NSF of Guangxi(2023JJA110104);Basic Ability Improvement Project for Middle-Aged and Young Teachers of Universities in Guangxi(2022KY0610);Innovation Project of Guangxi Graduate Education(YCBZ2023021)

摘要/Abstract

摘要：

该文研究了闵可夫斯基空间中矩阵的 $\mathfrak{m}$-WG 逆的性质和计算. 首先,利用值域和零空间给出了 $\mathfrak{m}$-WG 逆的刻画.其次, 给出了 $\mathfrak{m}$-WG 逆与非奇异加边矩阵之间的关系, 并讨论了 $\mathfrak{m}$-WG 逆的扰动界. 最后, 利用逐次矩阵平方算法给出了 $\mathfrak{m}$-WG 逆的计算.

关键词: $\mathfrak{m}$-WG 逆, 加边矩阵, 扰动界, 逐次矩阵平方算法

Abstract:

In this paper, the properties and computations of the $\mathfrak{m}$-WG inverse in Minskowski space are presented. Firstly, the characterization of the $\mathfrak{m}$-WG inverse is given by using the range and null space. Secondly, the relationship between the $\mathfrak{m}$-WG inverse and an invertible bordered matrix is given. Moreover, the perturbation bounds of the $\mathfrak{m}$-WG inverse is discussed. Finally, the successive matrix squaring algorithm is used to compute the $\mathfrak{m}$-WG inverse.

Key words: $\mathfrak{m}$-WG inverse, Bordered matrix, Perturbation bounds, Successive matrix squaring algorithm

中图分类号:

O151.2

韦华全, 吴辉, 刘晓冀, 靳宏伟. $\mathfrak{m}$-WG 逆的性质和计算[J]. 数学物理学报, 2024, 44(3): 547-562.

Wei Huaquan, Wu Hui, Liu Xiaoji, Jin Hongwei. Properties and Computations of the $\mathfrak{m}$-WG Inverse[J]. Acta mathematica scientia,Series A, 2024, 44(3): 547-562.

参考文献

[1]	Kyrchei I. Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations. Linear Algebra Appl, 2013, 438(1): 136-152
[2]	Diao H A, Wei Y M, Xie P P. Small sample statistical condition estimation for the total least squares problem. Numer Algoritm, 2017, 5(2): 435-455
[3]	Donchenko V S, Kirichenko N F. Generalized inverse in control with constraints. Cybern Syst Anal, 2003, 39(6): 854-861
[4]	Campbell S L, Meyer C D, Rose N J. Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients. SIAM J Appl Math, 1976, 31(3): 411-425
[5]	Meyer C D, Shoaf J M. Updating finite Markov chains by using techniques of group matrix inversion. J Stat Comput Sim, 1980, 11(3/4): 163-181
[6]	Ren B Y, Wang Q W, Chen X Y. The $\eta$-anti-Hermitian solution to a system of constrained matrix equations over the generalized Segre quaternion algebra. Symmetry, 2023, 15(3): 592
[7]	Chen X Y, Wang Q W. The $\eta$-(anti-) Hermitian solution to a constrained Sylvester-type generalized commutative quaternion matrix equation. Banach J Math Anal, 2023, 17: Article number 40
[8]	Xu X L, Wang Q W. The consistency and the general common solution to some quaternion matrix equations. Ann Funct Anal, 2023, 14(3): 53
[9]	Brock K G. A note on commutativity of a linear operatorand its Moore-Penrose inverse. Numer Func Anal Opt, 1990, 11(7/8): 673-678
[10]	Rako? evi? V. Moore-Penrose inverse in Banach algebras. Proc R Ir Acad Ser, 1988, 88(1): 57-60
[11]	Meenakshi A R. Generalized inverses of matrices in Minkowski space. Proc Nat Seminar Alg Appl, 2000, 1: 1-14
[12]	Ramon C, Rauscher E A. Superluminal transformations in complex Minkowski spaces. Found Phys, 1980, 10(7/8): 661-669
[13]	Renardy M. Singular value decomposition in Minkowski space. Linear Algebra Appl, 1996, 236: 53-58
[14]	Xing Z F. On the deterministic and non-deterministic Mueller matrix. J Mod Optic, 1992, 39(3): 461-484
[15]	K?l??cman A, Al-Zhour Z. The representation and approximation for the weighted Minkowski inverse in Minkowski space. Math Comput Model, 2008, 47(3/4): 363-371
[16]	Al-Zhour Z. Extension and generalization properties of the weighted Minkowski inverse in a Minkowski space for an arbitrary matrix. Comput Math Appl, 2015, 70(5): 954-961
[17]	Meenakshi A R, Krishnaswamy D. On sums of range symmetric matrices in Minkowski space. B Malays Math Sci So, 2002, 25(2): 137-148
[18]	Meenakshi A R, Krishnaswamy D. Product of range symmetric block matrices in Minkowski space. B Malays Math Sci So, 2006, 29(1): 59-68
[19]	Wang H X, Li N, Liu X J. The $\mathfrak{m}$-core inverse and its applications. Linear Multilinear A, 2021, 69(13): 2491-2509
[20]	Wang H X, Wu H, Liu X J. The $\mathfrak{m}$-core-EP inverse in Minkowski space. B Iran Math Soc, 2022, 48(5): 2577-2601
[21]	Wang H X, Chen J L. Weak group inverse. Open Math, 2018, 16(1): 1218-1232
[22]	Prasad K M, Mohana K S. Core-EP inverse. Linear Multilinear A, 2014, 62(6): 792-802
[23]	Wang H X, Liu X J. The weak group matrix. Aequationes Math, 2019, 93(7): 1261-1273
[24]	Ferreyra D E, Orquera V, Thome N. A weak group inverse for rectangular matrices. RACSAM Rev R Acad A, 2019, 113(4): 3727-3740
[25]	Zhou M M, Chen J L, Zhou Y K. Weak group inverses in proper *-rings. J Algebra Appl, 2020, 19(12): 2050238
[26]	Xu S Z, Wang H X, Chen J L, et al. Generalized WG inverse. J Algebra Appl, 2021, 20(5): 2150072
[27]	Mosi? D, Stanimirovi? P S. Representations for the weak group inverse. Appl Math Comput, 2021, 397(6): 125957
[28]	Yan H, Wang H X, Zuo K Z, et al. Further characterizations of the weak group inverse of matrices and the weak group matrix. AIMS Math, 2021, 6(9): 9322-9341
[29]	Zhou M M, Chen J L, Zhou Y K, et al. Weak group inverses and partial isometries in proper $\ast$-rings. Linear Multilinear A, 2022, 70(19): 4528-4543
[30]	Mosi′c D, Zhang D C. Weighted weak group inverse for Hilbert space operators. Front Math China, 2020, 15(6): 709-726
[31]	Ferreyra D E, Orquera V, Thome N. Representations of the weighted WG inverse and a rank equation’s solution. Linear Multilinear A, 2023, 71(2): 226-241
[32]	Zhou Y K, Chen J L, Zhou M M. $m$-weak group inverses in a ring with involution. RACSAM Rev R Acad A, 2021, 115(1): 1-13
[33]	Wu H, Wang H X, Jin H W. The $\mathfrak{m}$-WG inverse in Minkowski space. Filomat, 2022, 36(4): 1125-1141
[34]	Wang H X. Core-EP decomposition and its applications. Linear Algebra Appl, 2016, 508: 289-300
[35]	Wang G R, Wei Y M, Qiao S Z. Generalized Inverses:Theory and Computations. Beijing: Science Press, 2018
[36]	Stewart G W. On the continuity of the generalized inverse. SIAM J Appl Math, 1969, 17(1): 33-45
[37]	Stanimirovi? P S, Cvetkovi? -Ili? D S. Successive matrix squaring algorithm for computing outer inverses. Appl Math Comput, 2008, 203(1): 19-29