Some new characterizations of the generalized Moore-Penrose inverse are proposed using range, null space, several matrix equations and projectors. Several representations of the generalized Moore-Penrose inverse are given. The relationships between the generalized Moore-Penrose inverse and other generalized inverses are discussed using core-EP decomposition. The generalized Moore-Penrose matrices are introduced and characterized. One relation between the generalized Moore-Penrose inverse and corresponding nonsingular border matrix is presented. In addition, applications of the generalized Moore-Penrose inverse in solving restricted matrix equations are studied.
Citation: Yang Chen, Kezheng Zuo, Zhimei Fu. New characterizations of the generalized Moore-Penrose inverse of matrices[J]. AIMS Mathematics, 2022, 7(3): 4359-4375. doi: 10.3934/math.2022242
Some new characterizations of the generalized Moore-Penrose inverse are proposed using range, null space, several matrix equations and projectors. Several representations of the generalized Moore-Penrose inverse are given. The relationships between the generalized Moore-Penrose inverse and other generalized inverses are discussed using core-EP decomposition. The generalized Moore-Penrose matrices are introduced and characterized. One relation between the generalized Moore-Penrose inverse and corresponding nonsingular border matrix is presented. In addition, applications of the generalized Moore-Penrose inverse in solving restricted matrix equations are studied.
| [1] | O. Baksalary, G. Trenkler, Core inverse of matrices, Linear Multilinear Algebra, 58 (2010), 681–697. http://dx.doi.org/10.1080/03081080902778222 |
| [2] | A. Ben-Israel, T. Greville, Generalized inverses: theory and applications, New-York: Springer-Verlag, 2003. http://dx.doi.org/10.1007/b97366 |
| [3] |
D. Cvetković-Ilić, C. Deng, Some results on the Drazin invertibility and idempotents, J. Math. Anal. Appl., 359 (2009), 731–738. http://dx.doi.org/10.1016/j.jmaa.2009.05.062 doi: 10.1016/j.jmaa.2009.05.062
|
| [4] | D. Cvetković-Ilić, Y. Wei, Algebraic properties of generalized inverses, Singapore: Springer Nature, 2017. http://dx.doi.org/10.1007/978-981-10-6349-7 |
| [5] |
M. Drazin, Pseudo-inverses in associative rings and semigroups, The American Mathematical Monthly, 65 (1958), 506–514. http://dx.doi.org/10.1080/00029890.1958.11991949 doi: 10.1080/00029890.1958.11991949
|
| [6] |
D. Ferreyra, F. Levis, N. Thome, Characterizations of k-commutative equalities for some outer generalized inverses, Linear Multilinear Algebra, 68 (2020), 177–192. http://dx.doi.org/10.1080/03081087.2018.1500994 doi: 10.1080/03081087.2018.1500994
|
| [7] |
D. Ferreyra, F. Levis, N. Thome, Revisiting the core-EP inverse and its extension to rectangular matrices, Quaest. Math., 41 (2018), 265–281. http://dx.doi.org/10.2989/16073606.2017.1377779 doi: 10.2989/16073606.2017.1377779
|
| [8] |
C. Hung, T. Markham, The Moore-Penrose inverse of a partioned matrix $M = \left[ {\begin{array}{*{20}{c}} A & B \\ C & D\\ \end{array}} \right]$, Linear Algebra Appl., 11 (1975), 73–86. http://dx.doi.org/10.1016/0024-3795(75)90118-4 doi: 10.1016/0024-3795(75)90118-4
|
| [9] |
H. Ma, X. Gao, P. Stanimirović, Characterizations, iterative method, sign pattern and perturbation analysis for the DMP inverse with its applications, Appl. Math. Comput., 378 (2020), 125196. http://dx.doi.org/10.1016/j.amc.2020.125196 doi: 10.1016/j.amc.2020.125196
|
| [10] |
H. Ma, P. Stanimirović, Characterizations, approximation and pertuibatins of the core-EP inverse, Appl. Math. Comput., 359 (2019), 404–417. http://dx.doi.org/10.1016/j.amc.2019.04.071 doi: 10.1016/j.amc.2019.04.071
|
| [11] |
S. Malik, N. Thome, On a new generalized inverse for matrices of an arbitrary index, Appl. Math. Comput., 226 (2014), 575–580. http://dx.doi.org/10.1016/j.amc.2013.10.060 doi: 10.1016/j.amc.2013.10.060
|
| [12] |
D. Mosić, P. Stanimirović, Representations for the weak group inverse, Appl. Math. Comput., 397 (2021), 125957. http://dx.doi.org/10.1016/j.amc.2021.125957 doi: 10.1016/j.amc.2021.125957
|
| [13] |
R. Penrose, A generalized inverse for matrices, Math. Proc. Cambridge, 51 (1955), 406–413. http://dx.doi.org/10.1017/S0305004100030401 doi: 10.1017/S0305004100030401
|
| [14] | K. Prasad, K. Mohana, Core-EP inverse, Linear Multilinear Algebra, 62 (2014), 792–802. http://dx.doi.org/10.1080/03081087.2013.791690 |
| [15] |
K. Stojanović, D. Mosić, Generalization of the Moore-Penrose inverse, RACSAM, 114 (2020), 196. http://dx.doi.org/10.1007/s13398-020-00928-x doi: 10.1007/s13398-020-00928-x
|
| [16] |
H. Wang, Core-EP decomposition and its applications, Linear Algebra Appl., 508 (2016), 289–300. http://dx.doi.org/10.1016/j.laa.2016.08.008 doi: 10.1016/j.laa.2016.08.008
|
| [17] | H. Wang, J. Chen, Weak group inverse, Open Math., 16 (2017), 1218–1232. http://dx.doi.org/10.1515/math-2018-0100 |
| [18] | H. Wang, X. Liu, The weak group matrix, Aequat. Math., 93 (2019), 1261–1273. http://dx.doi.org/10.1007/s00010-019-00639-8 |
| [19] |
H. Yan, H. Wang, K. Zuo, Y. Chen, Further characterizations of the weak group inverse of matrices and the weak group matrix, AIMS Mathematics, 6 (2021), 9322–9341. http://dx.doi.org/10.3934/math.2021542 doi: 10.3934/math.2021542
|
| [20] |
K. Zuo, O. Baksalary, D. Cvetković-Ilić, Further characterizations of the co-EP matrices, Linear Algebra Appl., 616 (2021), 66–83. http://dx.doi.org/10.1016/j.laa.2020.12.029 doi: 10.1016/j.laa.2020.12.029
|
| [21] |
K. Zuo, Y. Cheng, The new revisitation of core-EP inverse of matrices, Filomat, 33 (2019), 3061–3072. http://dx.doi.org/10.2298/FIL1910061Z doi: 10.2298/FIL1910061Z
|
| [22] | K. Zuo, D. Cvetković-Ilić, Y. Cheng, Different characterizations of DMP-inverse of matrices, Linear Multilinear Algebra, (2020), in press. http://dx.doi.org/10.1080/03081087.2020.1729084 |
Yang Chen, Kezheng Zuo, Zhimei Fu. New characterizations of the generalized Moore-Penrose inverse of matrices[J]. AIMS Mathematics, 2022, 7(3): 4359-4375. doi: 10.3934/math.2022242
DownLoad: