In this paper, we introduce certain different characterizations and several new properties of the $ m $-weak group inverse of a complex matrix. Also, the relationship between the $ m $-weak group inverse and a nonsingular bordered matrix is established as well as the Cramer's rule for the solution of the restricted matrix equation that depends on the $ m $-weak group inverse.
Citation: Wanlin Jiang, Kezheng Zuo. Further characterizations of the $ m $-weak group inverse of a complex matrix[J]. AIMS Mathematics, 2022, 7(9): 17369-17392. doi: 10.3934/math.2022957
In this paper, we introduce certain different characterizations and several new properties of the $ m $-weak group inverse of a complex matrix. Also, the relationship between the $ m $-weak group inverse and a nonsingular bordered matrix is established as well as the Cramer's rule for the solution of the restricted matrix equation that depends on the $ m $-weak group inverse.
| [1] |
M. P. Drazin, Pseudo-inverses in associative rings and semigroups, Am. Math. Mon., 65 (1958), 506–514. https://doi.org/10.2307/2308576 doi: 10.2307/2308576
|
| [2] |
Y. Wei, G. Wang, The perturbation theory for the Drazin inverse and its applications, Linear Algebra Appl., 258 (1997), 179–186. https://doi.org/10.1016/s0024-3795(96)00159-0 doi: 10.1016/s0024-3795(96)00159-0
|
| [3] |
R. E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl., 322 (2001), 207–217. https://doi.org/10.1016/j.amc.2009.05.021 doi: 10.1016/j.amc.2009.05.021
|
| [4] | D. S. Cvetkovi$\rm \acute{c}$-Ili$\rm\acute{c}$, Xiaoji Liu, Y. Wei, Some additive results for the generalized Drazin inverse in a Banach algebra, Electronic J. Linear Algebra, 22 (2011), 1049–1058. |
| [5] |
D. S. Cvetkovi$\rm \acute{c}$-Ili$\rm\acute{c}$, New additive results on Drazin inverse and its applications, Appl. Math. Comp., 218 (2011), 3019–3024. https://doi.org/10.1016/j.amc.2011.08.083 doi: 10.1016/j.amc.2011.08.083
|
| [6] |
N. P. Karampetakis, S. Vologiannidis, DFT calculation of the generalized and Drazin inverse of a polynomial matrix, Appl. Math. Comput., 143 (2003), 501–521. https://doi.org/10.1016/S0096-3003(02)00377-6 doi: 10.1016/S0096-3003(02)00377-6
|
| [7] |
N. P. Karampetakis, P. S. Stanimirovi$\rm \acute{c}$, M. B. Tasi$\rm \acute{c}$, On the computation of the Drazin inverse of a polynomial matrix, Far East J. Math. Sci., 26 (2007), 1–24. https://doi.org/10.1016/s1474-6670(17)38994-2 doi: 10.1016/s1474-6670(17)38994-2
|
| [8] |
P. S. Stanimirovi$\rm \acute{c}$, V. N. Katsikis, H. Ma, Representations and properties of the W-Weighted Drazin inverse, Linear Multilinear A., 65 (2017), 1080–1096. https://doi.org/10.1080/03081087.2016.1228810 doi: 10.1080/03081087.2016.1228810
|
| [9] |
I. I. Kyrchei, Weighted quaternion core-EP, DMP, MPD, and CMP inverses and their determinantal representations, RACSAM Rev. R. Acad. A, 114 (2020), 198. https://doi.org/10.1007/s13398-020-00930-3 doi: 10.1007/s13398-020-00930-3
|
| [10] |
S. B. Malik, N. Thome, On a new generalized inverse for matrices of an arbitrary index, Appl. Math. Comput., 226 (2014), 575–580. https://doi.org/10.1016/j.amc.2013.10.060 doi: 10.1016/j.amc.2013.10.060
|
| [11] |
K. Zuo, D. S. Cvetkovi$\rm \acute{c}$-Ili$\rm\acute{c}$, Y. Cheng, Different characterizations of DMP-inverse of matrices, Linear Multilinear A., 70 (2022), 411–418. https://doi.org/10.1080/03081087.2020.1729084 doi: 10.1080/03081087.2020.1729084
|
| [12] |
L. Meng, The DMP inverse for rectangular matrices, Filomat, 31 (2017), 6015–6019. https://doi.org/10.2298/FIL1719015M doi: 10.2298/FIL1719015M
|
| [13] |
I. I. Kyrchei, D. Mosi$\rm \acute{c}$, P. S. Stanimirovi$\rm \acute{c}$, MPD-DMP-solutions to quaternion two-sided restricted matrix equations, Comput. Appl. Math., 40 (2021), 177. https://doi.org/10.1007/s40314-021-01566-8 doi: 10.1007/s40314-021-01566-8
|
| [14] | D. S. Cvetkovi$\rm \acute{c}$-Ili$\rm\acute{c}$, Y. Wei, Algebraic properties of generalized inverses, In: Developments on Mathematics, Springer, Singapore, 52 (2017). http://dx.doi.org/10.1007/978-981-10-6349-7 |
| [15] | H. Wang, J. Chen, Weak group inverse, Open Math., 16 (2018), 1218–1232. https://doi.org/10.1515/math-2018-0100 |
| [16] |
H. Wang, Core-EP decomposition and its applications, Linear Algebra Appl., 508 (2016), 289–300. https://doi.org/10.1016/j.laa.2016.08.008 doi: 10.1016/j.laa.2016.08.008
|
| [17] | H. Wang, X. Liu, The weak group matrix, Aequationes Math., 93 (2019), 1261–1273. https://doi.org/10.1007/s00010-019-00639-8 |
| [18] | M. Zhou, J. Chen, Y. Zhou, Weak group inverses in proper *-rings, J. Algebra Appl., 19 (2020). https://doi.org/10.1142/S0219498820502382 |
| [19] | Y. Zhou, J. Chen, M. Zhou, $m$-weak group inverses in a ring with involution, RACSAM Rev. R. Acad. A, 115 (2021). https://doi.org/10.1007/s13398-020-00932-1 |
| [20] | M. Zhou, J. Chen, Y. Zhou, N. Thome, Weak group inverses and partial isometries in proper *-rings, Linear Multilinear A., 2021. https://doi.org/10.1080/03081087.2021.1884639 |
| [21] |
D. E. Ferreyra, V. Orquera, N. Thome, A weak group inverse for rectangular matrices, RACSAM Rev. R. Acad. A, 113 (2019), 3727–3740. https://doi.org/10.1007/s13398-019-00674-9 doi: 10.1007/s13398-019-00674-9
|
| [22] | Y. Zhou, J. Chen, Weak core inverses and pseudo core inverses in a ring with involution, Linear Multilinear A., 2021. https://doi.org/10.1080/03081087.2021.1971151 |
| [23] |
R. Penrose, A generalized inverse for matrices, Math. Proc. Cambridge Philos. Soc., 51 (1955), 406–413. https://doi.org/10.1017/S0305004100030401 doi: 10.1017/S0305004100030401
|
| [24] | G. Wang, Y. Wei, S. Qiao, Generalized inverses: Theory and computations, In: Developments on Mathematics, Springer and Beijing, Science Press, 53 (2018). http://dx.doi.org/10.1007/978-981-13-0146-9 |
| [25] | O. M. Baksalary, G. Trenkler, Core inverse of matrices, Linear Multilinear A., 58 (2010), 681–697. http://dx.doi.org/10.1080/03081080902778222 |
| [26] |
H. Kurata, Some theorems on the core inverse of matrices and the core partial ordering, Appl. Math. Comput., 316 (2018), 43–51. http://dx.doi.org/10.1016/j.amc.2017.07.082 doi: 10.1016/j.amc.2017.07.082
|
| [27] | G. Luo, K. Zuo, L. Zhou, Revisitation of core inverse, Wuhan Univ. J. Nat. Sci., 20 (2015), 381–385. https: //doi.org/10.1007/s11859-015-1109-6 |
| [28] | K. Prasad, K. S Mohana, Core-EP inverse, Linear Multilinear A., 62 (2014), 792–802. http://dx.doi.org/10.1080/03081087.2013.791690 |
| [29] |
D. E. Ferreyra, F. E. Levis, N. Thome, Revisiting the core-EP inverse and its extension to rectangular matrices, Quaest. Math., 41 (2018), 1–17. http://dx.doi.org/10.2989/16073606.2017.1377779 doi: 10.2989/16073606.2017.1377779
|
| [30] |
K. Zuo, Y. Cheng, The new revisitation of core EP inverse of matrices, Filomat, 33 (2019), 3061–3072. https://doi.org/10.2298/FIL1910061Z doi: 10.2298/FIL1910061Z
|
| [31] | J. Benítez, E. Boasso, H. Jin, On one-sided $(B, C)$-inverse of arbitrary matrices, Electron. J. Linear Al., 32 (2017), 391–422. https://arXiv.org/pdf/1701.09054v1.pdf |
| [32] |
M. P. Drazin, A class of outer generalized inverses, Linear Algebra Appl., 436 (2012), 1909–1923. https://doi.org/10.1016/j.laa.2011.09.004 doi: 10.1016/j.laa.2011.09.004
|
| [33] |
X. Wang, C. Deng, Properties of $m$-EP operators, Linear Multilinear A., 65 (2017), 1349–1361. https://doi.org/10.1080/03081087.2016.1235131 doi: 10.1080/03081087.2016.1235131
|
| [34] |
C. Deng, H. Du, Representation of the Moore-Penrose inverse of $2\times2$ block operator valued matrices, J. Korean Math. Soc., 46 (2009), 1139–1150. https://doi.org/10.4134/JKMS.2009.46.6.1139 doi: 10.4134/JKMS.2009.46.6.1139
|
| [35] |
D. E. Ferreyra, F. E. Levis, N. Thome, Maximal classes of matrices determining generalized inverses, Appl. Math. Comput., 333 (2018), 42–52. https://doi.org/10.1016/j.amc.2018.03.102 doi: 10.1016/j.amc.2018.03.102
|
| [36] |
D. E. Ferreyra, F. E. Levis, N. Thome, Characterizations of $k$-commutative equalities for some outer generalized inverses, Linear Multilinear A., 68 (2020), 177–192. https://doi.org/10.1080/03081087.2018.1500994 doi: 10.1080/03081087.2018.1500994
|
| [37] |
Y. Yuan, K. Zuo, Compute $\lim_{\lambda \to 0}X(\lambda I_{p}+YAX)^{-1}Y$ be the product singular value decomposition, Linear Multilinear A., 64 (2016), 269–278. https://doi.org/10.1080/03081087.2015.1034641 doi: 10.1080/03081087.2015.1034641
|
| [38] | P. S. Stanimirovi$\rm \acute{c}$, M. $\rm \acute{S}$iri$\rm\acute{c}$, I. Stojanovi$\rm \acute{c}$, D. Gerontitis, Conditions for existence, representations, and computation of matrix generalized inverses, Complexity, 2017 (2017). https://doi.org/10.1155/2017/6429725 |
| [39] | Y. Wei, P. S. Stanimirovi$\rm \acute{c}$, M. D. Petkovi$\rm \acute{c}$, Numerical and symbolic computations of generalized inverses, World Scientific Publishing Co. Pte. Ltd., Hackensack, 2018. http://dx.doi.org/10.1142/10950 |
| [40] |
P. S. Stanimirovi$\rm \acute{c}$, M. D. Petkovi$\rm \acute{c}$, D. Mosi$\rm\acute{c}$, Exact solutions and convergence of gradient based dynamical systems for computing outer inverses, Appl. Math. Comput., 412 (2022), 126588. https://doi.org/10.1016/j.amc.2021.126588 doi: 10.1016/j.amc.2021.126588
|
Wanlin Jiang, Kezheng Zuo. Further characterizations of the $ m $-weak group inverse of a complex matrix[J]. AIMS Mathematics, 2022, 7(9): 17369-17392. doi: 10.3934/math.2022957
DownLoad: