New characterizations for generalized inverses along the core parts of three matrix decompositions were investigated in this paper. Let $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $ be the core parts of the core-nilpotent decomposition, the core-EP decomposition and EP-nilpotent decomposition of $ A\in \mathbb{C}^{n\times n} $, respectively, where EP denotes the EP matrix. A number of characterizations and different representations of the Drazin inverse, the weak group inverse and the core-EP inverse were given by using the core parts $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $. One can prove that, the Drazin inverse is the inverse along $ A_{1} $, the weak group inverse is the inverse along $ \hat{A}_{1} $ and the core-EP inverse is the inverse along $ \tilde{A}_{1} $. A unified theory presented in this paper covers the Drazin inverse, the weak group inverse and the core-EP inverse based on the core parts of the core-nilpotent decomposition, the core-EP decomposition and EP-nilpotent decomposition of $ A\in \mathbb{C}^{n\times n} $, respectively. In addition, we proved that the Drazin inverse of $ A $ is the inverse of $ A $ along $ U $ and $ A_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $; the weak group inverse of $ A $ is the inverse of $ A $ along $ U $ and $ \hat{A}_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $; the core-EP inverse of $ A $ is the inverse of $ A $ along $ U $ and $ \tilde{A}_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $. Let $ X_{1} $, $ X_{4} $ and $ X_{7} $ be the generalized inverses along $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $, respectively. In the last section, some useful examples were given, which showed that the generalized inverses $ X_{1} $, $ X_{4} $ and $ X_{7} $ were different generalized inverses. For a certain singular complex matrix, the Drazin inverse coincides with the weak group inverse, which is different from the core-EP inverse. Moreover, we showed that the Drazin inverse, the weak group inverse and the core-EP inverse can be the same for a certain singular complex matrix.
Citation: Xiaofei Cao, Yuyue Huang, Xue Hua, Tingyu Zhao, Sanzhang Xu. Matrix inverses along the core parts of three matrix decompositions[J]. AIMS Mathematics, 2023, 8(12): 30194-30208. doi: 10.3934/math.20231543
New characterizations for generalized inverses along the core parts of three matrix decompositions were investigated in this paper. Let $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $ be the core parts of the core-nilpotent decomposition, the core-EP decomposition and EP-nilpotent decomposition of $ A\in \mathbb{C}^{n\times n} $, respectively, where EP denotes the EP matrix. A number of characterizations and different representations of the Drazin inverse, the weak group inverse and the core-EP inverse were given by using the core parts $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $. One can prove that, the Drazin inverse is the inverse along $ A_{1} $, the weak group inverse is the inverse along $ \hat{A}_{1} $ and the core-EP inverse is the inverse along $ \tilde{A}_{1} $. A unified theory presented in this paper covers the Drazin inverse, the weak group inverse and the core-EP inverse based on the core parts of the core-nilpotent decomposition, the core-EP decomposition and EP-nilpotent decomposition of $ A\in \mathbb{C}^{n\times n} $, respectively. In addition, we proved that the Drazin inverse of $ A $ is the inverse of $ A $ along $ U $ and $ A_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $; the weak group inverse of $ A $ is the inverse of $ A $ along $ U $ and $ \hat{A}_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $; the core-EP inverse of $ A $ is the inverse of $ A $ along $ U $ and $ \tilde{A}_{1} $ for any $ U\in \{A_{1}, \hat{A}_{1}, \tilde{A}_{1}\} $. Let $ X_{1} $, $ X_{4} $ and $ X_{7} $ be the generalized inverses along $ A_{1} $, $ \hat{A}_{1} $ and $ \tilde{A}_{1} $, respectively. In the last section, some useful examples were given, which showed that the generalized inverses $ X_{1} $, $ X_{4} $ and $ X_{7} $ were different generalized inverses. For a certain singular complex matrix, the Drazin inverse coincides with the weak group inverse, which is different from the core-EP inverse. Moreover, we showed that the Drazin inverse, the weak group inverse and the core-EP inverse can be the same for a certain singular complex matrix.
| [1] |
J. Benítez, E. Boasso, H. W. Jin, On one-sided $(B, C)$-inverses of arbitrary matrices, Electron. J. Linear Al., 32 (2017), 391–422. https://doi.org/10.13001/1081-3810.3487 doi: 10.13001/1081-3810.3487
|
| [2] |
R. B. Bapat, S. K. Jain, K. M. P. Karantha, M. D. Raj, Outer inverses: characterization and application, Linear Algebra Appl., 528 (2017), 171–184. https://doi.org/10.1016/j.laa.2016.06.045 doi: 10.1016/j.laa.2016.06.045
|
| [3] |
E. Boasso, G. Kantún-Montiel, The $(b, c)$-inverses in rings and in the Banach context, Mediterr. J. Math., 14 (2017), 112. http://doi.org/ 10.1007/s00009-017-0910-1 doi: 10.1007/s00009-017-0910-1
|
| [4] |
O. M. Baksalary, G. Trenkler, Core inverse of matrices, Linear Multilinear A., 58 (2010), 681–697. http://doi.org/10.1080/03081080902778222 doi: 10.1080/03081080902778222
|
| [5] | S. L. Campbell, C. D. Meyer, Generalized inverses of linear transformations, Philadelphia: SIAM, 2009. https://doi.org/10.1137/1.9780898719048 |
| [6] | X. F. Cao, S. Z. Xu, X. C. Wang, K. Liu, Two generalized constrained inverses based on the core part of the core-EP decomposition of a complex matrix, ScienceAsia, (accepted). |
| [7] |
M. P. Drazin, Pseudo-inverses in associative rings and semigroup, Am. Math. Mon., 65 (1958), 506–514. http://doi.org/10.1080/00029890.1958.11991949 doi: 10.1080/00029890.1958.11991949
|
| [8] |
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
|
| [9] | J. A. Green, On the structure of semigroups, Ann. Math., 54 (1951), 163–172. |
| [10] |
R. E. Hartwig, Block generalized inverses, Arch. Rational Mech. Anal., 61 (1976), 197–251. https://doi.org/ 10.1007/BF00281485 doi: 10.1007/BF00281485
|
| [11] |
R. E. Hartwig, K. Spindelböck, Matrices for which $A^{\ast}$ and $A^{\dagger}$ commmute, Linear Multilinear A., 14 (1983), 241–256. https://doi.org/10.1080/03081088308817561 doi: 10.1080/03081088308817561
|
| [12] |
X. Mary, On generalized inverse and Green's relations, Linear Algebra Appl., 434 (2011), 1836–1844. https://doi.org/10.1016/j.laa.2010.11.045 doi: 10.1016/j.laa.2010.11.045
|
| [13] | E. H. Moore, On the reciprocal of the general algebraic matrix, B. Am. Math. Soc., 26 (1920), 394–395. |
| [14] | S. K. Mitra, P. Bhimasankaram, S. B. Malik, Matrix partial orders, shorted operators and applications, Singaproe: Word Scientific, 2010. https://doi.org/10.1142/7170 |
| [15] |
K. M. Prasad, K. S. Mohana, Core-EP inverse, Linear Multilinear A., 62 (2014), 792–804. https://doi.org/10.1080/03081087.2013.791690 doi: 10.1080/03081087.2013.791690
|
| [16] |
X. Mary, P. Patrício, Generalized inverses modulo $\mathcal{H}$ in semigroups and rings, Linear Multilinear A., 61 (2013), 1130–1135. https://doi.org/10.1080/03081087.2012.731054 doi: 10.1080/03081087.2012.731054
|
| [17] |
R. Penrose, A generalized inverse for matrices, Math. Proc. Cambridge, 51 (1955), 406–413. http://doi.org/10.1017/S0305004100030401 doi: 10.1017/S0305004100030401
|
| [18] | C. R. Rao, S. K. Mitra, Generalized inverse of matrices and its applications, New York: Wiley, 1971. |
| [19] |
D. S. Rakić, A note on Rao and Mitra's constrained inverse and Drazin's (b, c) inverse, Linear Algebra Appl., 523 (2017), 102–108. https://doi.org/10.1016/j.laa.2017.02.025 doi: 10.1016/j.laa.2017.02.025
|
| [20] | C. R. Rao, S. K. Mitra, Generalized inverse of a matrix and its application, In: Theory of statistics, Berkeley: University of California Press, 1972,601–620. https://doi.org/10.1525/9780520325883-032 |
| [21] |
H. X. 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
|
| [22] |
H. X. Wang, J. L. Chen, Weak group inverse, Open Math., 16 (2018), 1218–1232. http://doi.org/10.1515/math-2018-0100 doi: 10.1515/math-2018-0100
|
| [23] |
H. X. Wang, X. J. Liu, EP-nilpotent decomposition and its applications, Linear Multilinear A., 68 (2020), 1682–1694. https://doi.org/10.1080/03081087.2018.1555571 doi: 10.1080/03081087.2018.1555571
|
| [24] |
S. Z. Xu, J. Benítez, Existence criteria and expressions of the $(b, c)$-inverse in rings and their applications, Mediterr. J. Math., 15 (2018), 14. https://doi.org/10.1007/s00009-017-1056-x doi: 10.1007/s00009-017-1056-x
|
Xiaofei Cao, Yuyue Huang, Xue Hua, Tingyu Zhao, Sanzhang Xu. Matrix inverses along the core parts of three matrix decompositions[J]. AIMS Mathematics, 2023, 8(12): 30194-30208. doi: 10.3934/math.20231543
DownLoad: