Solving an open problem about the G-Drazin partial order

[EN] G-Drazin inverses and the G-Drazin partial order for square matrices have been both recently introduced by Wang and Liu. They proved the following implication: If A is below B under the G-Drazin partial order, then any G-Drazin inverse of B is also a G-Drazin inverse of A. However, this necessary condition could not be stated as a characterization and the validity (or not) of the converse implication was posed as an open problem. In this paper, this problem is completely solved. It is obtained that the converse, in general, is false, and a form to construct counterexamples is provided. It is also proved that the converse holds under an additional condition (which is also necessary) as well as for some special cases of matrices.Partially supported by Universidad Nacional de Río Cuarto (Grant PPI 18/C472), CONICET (Grant PIP 112-201501-00433CO), and by ANPCyT (Grant PICT 2018-03492) Partially supported by Universidad Nacional de La Pampa, Facultad de Ingeniería (Grant Resol. Nro. 155/14) Partially supported by Ministerio de Economía, Industria y Competitividad of Spain (Grant Red de Excelencia MTM2017-90682-REDT), and by Universidad Nacional del Sur of Argentina (Grant 24/L108)Ferreyra, DE.; Lattanzi, M.; Levis, FE.; Thome, N. (2020). Solving an open problem about the G-Drazin partial order. The Electronic Journal of Linear Algebra. 36:55-66. http://hdl.handle.net/10251/161871S55663

Representations of the weighted WG inverse and a rank equation's solution

In this paper, we present several representations of the W-weighted WG inverse. These representations are expressed in terms of matrix powers as well as in terms of matrix products involving only the Moore–Penrose inverse. In addition, a new characterization of the W-weighted WG inverse is presented by using a rank equation.Fil: Ferreyra, David Eduardo. Universidad Nacional de Río Cuarto. Facultad de Ciencias Exactas, Físico-Químicas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Orquera, Valentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Río Cuarto. Facultad de Ciencias Exactas, Físico-Químicas y Naturales. Departamento de Matemática; ArgentinaFil: Thome Coppo, Néstor Javier. Universidad Politécnica de Valencia; Españ

The weak core inverse

[EN] In this paper, we introduce a new generalized inverse, called weak core inverse (or, in short, WC inverse) of a complex square matrix. This new inverse extends the notion of the core inverse defined by Baksalary and Trenkler (Linear Multilinear Algebra 58(6):681-697, 2010). We investigate characterizations, representations, and properties for this generalized inverse. In addition, we introduce weak core matrices (or, in short, WC matrices) and we show that these matrices form a more general class than that given by the known weak group matrices, recently investigated by H. Wang and X. Liu.In what follows, we detail the acknowledgements. D.E. Ferreyra, F.E. Levis, A.N. Priori - Partially supported by Universidad Nacional de Rio Cuarto (Grant PPI 18/C559) and CONICET (Grant PIP 112-201501-00433CO). D.E. Ferreyra F.E. Levis - Partially supported by ANPCyT (Grant PICT 201803492). D.E. Ferreyra, N. Thome -Partially supported by Universidad Nacional de La Pampa, Facultad de Ingenieria (Grant Resol. Nro. 135/19). N. Thome -Partially supported by Ministerio de Economia, Industria y Competitividad of Spain (Grant Red de Excelencia MTM2017-90682-REDT) and by Universidad Nacional del Sur of Argentina (Grant 24/L108). We would like to thank the Referees for their valuable comments and suggestions which helped us to considerably improve the presentation of the paperFerreyra, DE.; Levis, FE.; Priori, AN.; Thome, N. (2021). The weak core inverse. Aequationes Mathematicae. 95(2):351-373. https://doi.org/10.1007/s00010-020-00752-zS351373952Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58(6), 681–697 (2010)Baksalary, O.M., Trenkler, G.: On a generalized core inverse. Appl. Math. Comput. 236(1), 450–457 (2014)Campbell, S.L., Meyer Jr., C.D.: Generalized Inverses of Linear Transformations. SIAM, Philadelphia (2009)Ceryan, N.: Handbook of Research on Trends and Digital Advances in Engineering Geology, Advances in Civil and Industrial Engineering. IGI Global, Hershey (2018)Chen, J.L., Mosić, D., Xu, S.Z.: On a new generalized inverse for Hilbert sapce operators. Quaest. Math. (2019). https://doi.org/10.2989/16073606.2019.1619104Cvetković-Ilić, D.S., Mosić, D., Wei, Y.: Partial orders on $$B(H)$$. Linear Algebra Appl. 481, 115–130 (2015)Djikić, M.S.: Lattice properties of the core-partial order. Banach J. Math. Anal. 11(2), 398–415 (2017)Doty, K.L., Melchiorri, C., Bonivento, C.: A theory of generalized inverses applied to robotics. Int. J. Robot. Res. 12(1), 1–19 (1993)Drazin, M.P.: Pseudo inverses in associative rings and semigroups. Am. Math. Mon. 65(7), 506–514 (1958)Ferreyra, D.E., Levis, F.E., Thome, N.: Revisiting of the core EP inverse and its extension to rectangular matrices. Quaest. Math. 41(2), 265–281 (2018)Ferreyra, D.E., Levis, F.E., Thome, N.: Maximal classes of matrices determining generalized inverses. Appl. Math. Comput. 333, 42–52 (2018)Ferreyra, D.E., Levis, F.E., Thome, N.: Characterizations of $$k$$-commutative equalities for some outer generalized inverses. Linear Multilinear Algebra 68(1), 177–192 (2020)Hartwig, R.E., Spindelböck, K.: Matrices for which $$A^*$$ and $$A^\dagger$$ conmmute. Linear Multilinear Algebra 14(3), 241–256 (1984)Liu, X., Cai, N.: High-order iterative methods for the DMP inverse. J. Math. Article ID 8175935, 6 p (2018)Malik, S., Thome, N.: On a new generalized inverse for matrices of an arbitrary index. Appl. Math. Comput. 226(1), 575–580 (2014)Malik, S., Rueda, L., Thome, N.: The class of $$m$$-EP and $$m$$-normal matrices. 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A weak group inverse for rectangular matrices

[EN] In this paper, we extend the notion of weak group inverse to rectangular matrices (called WweightedWGinverse) by using the weighted core EP inverse recently investigated. This new generalized inverse also generalizes the well-known weighted group inverse given by Cline and Greville. In addition, we give several representations of the W-weighted WG inverse, and derive some characterizations and properties.First author was partially supported by UNRC (Grant PPI 18/C472) and CONICET (Grant PIP 112-201501-00433CO). Third author was partially supported by Ministerio de Economia, Industria y Competitividad of Spain (Grants DGI MTM2013-43678-P and Red de Excelencia MTM2017-90682-REDT).Ferreyra, DE.; Orquera, V.; Thome, N. (2019). A weak group inverse for rectangular matrices. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 113(4):3727-3740. https://doi.org/10.1007/s13398-019-00674-9S372737401134Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58, 681–697 (2010)Baksalary, O.M., Trenkler, G.: On a generalized core inverse. Appl. Math. Comput. 236, 450–457 (2014)Bajodah, A.H.: Servo-constraint generalized inverse dynamics for robot manipulator control design. Int. J. Robot. Autom. 25, (2010). https://doi.org/10.2316/Journal.206.2016.1.206-3291Campbell, S.L., Meyer Jr., C.D.: Generalized Inverses of Linear transformations. SIAM, Philadelphia (2009)Cline, R.E., Greville, T.N.E.: A Drazin inverse for rectangular matrices. Linear Algebra Appl. 29, 53–62 (1980)Dajić, A., Koliha, J.J.: The weighted g-Drazin inverse for operators. J. Aust. Math. Soc. 2, 163–181 (2007)Doty, K.L., Melchiorri, C., Bonivento, C.: A theory of generalized inverses applied to robotics. Int. J. Rob. Res. 12, 1–19 (1993)Drazin, M.P.: Pseudo-inverses in associate rings and semirings. Am. Math. Mon. 65, 506–514 (1958)Ferreyra, D.E., Levis, F.E., Thome, N.: Revisiting of the core EP inverse and its extension to rectangular matrices. Quaest. Math. 41, 265–281 (2018)Ferreyra, D.E., Levis, F.E., Thome, N.: Maximal classes of matrices determining generalized inverses. Appl. Math. Comput. 333, 42–52 (2018)Gigola, S., Lebtahi, L., Thome, N.: The inverse eigenvalue problem for a Hermitian reflexive matrix and the optimization problem. J. Comput. Appl. Math. 291, 449–457 (2016)Hartwig, R.E.: The weighted $$*$$ ∗ -core-nilpotent decomposition. Linear Algebra Appl. 211, 101–111 (1994)Kirkland, S.J., Neumann, M.: Group inverses of M-matrices and their applications. Chapman and Hall/CRC, London (2013)Malik, S., Thome, N.: On a new generalized inverse for matrices of an arbitrary index. Appl. Math. Comput. 226, 575–580 (2014)Male $${{\check{\rm s}}}$$ s ˇ ević, B., Obradović, R., Banjac, B., Jovović, I., Makragić, M.: Application of polynomial texture mapping in process of digitalization of cultural heritage. arXiv:1312.6935 (2013). Accessed 14 June 2018Manjunatha Prasad, K., Mohana, K.S.: Core EP inverse. Linear Multilinear Algebra 62, 792–802 (2014)Mehdipour, M., Salemi, A.: On a new generalized inverse of matrices. Linear Multilinear Algebra 66, 1046–1053 (2018)Meng, L.S.: The DMP inverse for rectangular matrices. Filomat 31, 6015–6019 (2017)Mosić, D.: The CMP inverse for rectangular matrices. Aequaetiones Math. 92, 649–659 (2018)Penrose, R.: A generalized inverse for matrices. Proc. Cambrid. Philos. Soc. 51, 406–413 (1955)Soleimani, F., Stanimirović, P.S., Soleymani, F.: Some matrix iterations for computing generalized inverses and balancing chemical equations. Algorithms 8, 982–998 (2015)Xiao, G.Z., Shen, B.Z., Wu, C.K., Wong, C.S.: Some spectral techniques in coding theory. Discrete Math. 87, 181–186 (1991)Wang, H.: Core-EP decomposition and its applications. Linear Algebra Appl. 508, 289–300 (2016)Wang, H., Chen, J.: Weak group inverse. Open Math. 16, 1218–1232 (2018)Wei, Y.: A characterization for the $$W$$ W -weighted Drazin inverse and a Crammer rule for the $$W$$ W -weighted Drazin inverse solution. Appl. Math. Comput. 125, 303–310 (2002