Maximal classes of matrices determining generalized inverses

[EN] This paper derives some further results on recent generalized inverses studied in the literature, namely core EP, DMP, and CMP inverses. Our main aim is to develop maximal classes of matrices for which their representations remain valid. (C) 2018 Elsevier Inc. All rights reserved.Partially supported by a CONICET's Posdoctoral Research Fellowship, UNRC (grant PPI 18/C472) and CONICET (grant PIP 112-201501-00433CO). Partially supported by Ministerio de Economia y Competitividad of Spain (Grants DGI MTM2013-43678-P and Red de Excelencia MTM2017-90682-REDT).Ferreyra, DE.; Levis, F.; Thome, N. (2018). Maximal classes of matrices determining generalized inverses. Applied Mathematics and Computation. 333:42-52. https://doi.org/10.1016/j.amc.2018.03.102S425233

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

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. Linear Multilinear Algebra 64(11), 2119–2132 (2016)Manjunatha Prasad, K., Mohana, K.S.: Core EP inverse. Linear Multilinear Algebra 62(6), 792–802 (2014)Mehdipour, M., Salemi, A.: On a new generalized inverse of matrices. Linear Multilinear Algebra 66(5), 1046–1053 (2018)Mitra, S.K., Bhimasankaram, P., Malik, S.: Matrix Partial Orders, Shorted Operators and Applications, Series in Algebra, vol. 10. World Scientific Publishing Co. Pte. Ltd., Singapore (2010)Mosić, D., Stanimirović, P.S.: Composite outer inverses for rectangular matrices. Quaest. Math. (2019). https://doi.org/10.2989/16073606.2019.1671526Penrose, R.: A generalized inverse for matrices. Math. Proc. Cambr. Philos. Soc. 51(3), 406–413 (1955)Rakić, D.S., Dincić, N.C., Djordjević, D.S.: Core inverse and core partial order of Hilbert space operators. Appl. Math. Comput. 244(1), 283–302 (2014)Soleimani, F., Stanimirović, P.S., Soleymani, F.: Some matrix iterations for computing generalized inverses and balancing chemical equations. Algorithms 8(4), 982–998 (2015)Tosić, M., Cvetković-Ilić, D.S.: Invertibility of a linear combination of two matrices and partial orderings. Appl. Math. Comput. 218(9), 4651–4657 (2012)Wang, X.: Core-EP decomposition and its applications. Linear Algebra Appl. 508(1), 289–300 (2016)Wang, H., Chen, J.: Weak group inverse. Open Math. 16(1), 1218–1232 (2018)Wang, H., Liu, X.: The weak group matrix. Aequ. Math. 93(6), 1261–1273 (2019)Xiao, G.Z., Shen, B.Z., Wu, C.K., Wong, C.S.: Some spectral techniques in coding theory. Discrete Math. 87(2), 181–186 (1991)Zhou, M., Chen, J., Stanimirović, P., Katsikis, V.N., Ma, H.: Complex varying-parameter Zhang neural networks for computing core and core-EP inverse. Neural Process. Lett. 51, 1299–1329 (2020)Zhu, H.: On DMP inverses and $$m$$-EP elements in rings. Linear Multilinear Algebra 67(4), 756–766 (2019)Zhu, H., Patrício, P.: Several types of one-sided partial orders in rings. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113, 3177–3184 (2019

From projectors to 1MP and MP1 generalized inverses and their induced partial orders

[EN] This paper deals with new generalized inverses of rectangular complex matrices, namely 1MP and MP1-inverses. They are constructed from oblique projectors represented by means of inner generalized inverses, by using an adequate equivalence relation, and then passing to the quotient set. We give characterizations and general expressions for 1MP and MP1-inverses. As applications, the binary relations induced for these new generalized inverses are proved to be partial orders.M. V. Hernández and M. B. Lattanzi: Partially supported by Universidad Nacional de La Pampa, Facultad de Ingeniería (Grant Resol. Nro. 135/19). N. Thome: Partially supported by Universidad Nacional de La Pampa, Facultad de Ingeniería (Grant Resol. Nro. 135/19), by Ministerio de Economía y Competitividad of Spain (grant Red de Excelencia MTM2017-90682-REDT), by Universidad Nacional del Sur of Argentina (Grant PGI 24/L108), and by Universidad Nacional de Río Cuarto (Grant Res. Nro. 083/2020).Hernández, MV.; Lattanzi, MB.; Thome, N. (2021). From projectors to 1MP and MP1 generalized inverses and their induced partial orders. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 115(3):1-13. https://doi.org/10.1007/s13398-021-01090-8S1131153Baksalary, O.M., Styan, G.P.H., Trenkler, G.: On a matrix decomposition of Hartwig and Spindelböck. Linear Algebra Appl. 430, 2798–2812 (2009)Baksalary, O.M., Trenkler, G.: Core inverse of matrices. Linear Multilinear Algebra 58(6), 681–697 (2010)Ben-Israel, A., Greville, T.N.E.: Generalized inverses: theory and applications, 2nd edn. Springer-Verlag, New York (2003)Campbell, S.L., Meyer, C.D.: Generalized inverses of linear transformation. SIAM, Philadelphia (2009)Campbell, S.L., Meyer, C.D.: Weak Drazin inverses. Linear Algebra Appl. 20, 167–178 (1978)Chen, J.L., Zhu, H.H., Patrício, P., Zhang, Y.L.: Characterizations and representations of core and dual core inverses. Can. Math. Bull. 60(2), 269–282 (2017)Coll, C., Lattanzi, M., Thome, N.: Weighted G-Drazin inverses and a new pre-order on rectangular matrices. Appl. Math. Comput. 317, 12–24 (2018)Cvetković-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)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)Ferreyra, D.E., Lattanzi, M., Levis, F.E., Thome, N.: Parametrized solutions $$X$$ of the system $$AXA=AEA$$ and $$A^kEAX= XAEA^k$$ for a matrix $$A$$ having index $$k$$. Electron. J. Linear Algebra 35, 503–510 (2019)Hartwig, R.E., Spindleböck, K.: Matrices for which $$A^*$$ and $$A^{\dagger }$$ commute. Linear Multilinear Algebra 14, 241–256 (1984)Kyrchei, I.: Determinantal representations of the $$W$$-weighted Drazin inverse over the quaternion skew field. Appl. Math. Comput. 264, 453–465 (2015)Liu, X., Cai, N.: High-order iterative methods for the DMP inverse. J. Math. (2018). https://doi.org/10.1155/2018/8175935. (Article ID 8175935)Malik, S.B., Thome, N.: On a new generalized inverse for matrices of an arbitrary index. Appl. Math. Comput. 226, 575–580 (2014)Mehdipour, M., Salemi, A.: On a new generalized inverse of matrices. Linear Multilinear Algebra 66(5), 1046–1053 (2018)Manjunatha Prasad, K., Mohana, K.S.: Core EP inverse. Linear Multilinear Algebra 62(6), 792–802 (2014)Mitra, S.K., Bhimasankaram, P., Malik, S.: Matrix partial orders, shorted operators and applications. World Scientific Publishing Company, New Jersey (2010)Mosić, D., Kolundzija, M.Z.: Weighted CMP inverse of an operator between Hilbert spaces. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. 113, 2155–2173 (2019)Rakić, D.S., Dinčić, N.C., Djordjević, D.S.: Group, Moore-Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl. 463, 115–133 (2014)Rao, C.R., Mitra, S.K.: Generalized inverse of matrices and its applications. John Wiley & Sons Inc., New York (1971)Soleimani, F., Stanimirović, P.S., Soleymani, F.: Some matrix iterations for computing generalized inverses and balancing chemical equations. Algorithms 8(4), 982–998 (2015)Wang, H., Liu, X.: Characterizations of the core inverse and the partial ordering. Linear Multilinear Algebra 63(9), 1829–1836 (2015)Wang, X., Liu, X.: Partial orders based on core-nilpotent decomposition. Linear Algebra Appl. 488, 235–248 (2016)Zhu, H.H., Patrício, P.: Several types of one-sided partial orders in rings. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM. (2019). https://doi.org/10.1007/s13398-019-00685-6Zhu, H.: On DMP inverses and $$m$$-EP elements in rings. Linear Multilinear Algebra 67(4), 1–11 (2018)Zhou, M., Chen, J., Stanimirovic, P.S., Katsikis, V.N., Ma, H.: Complex varying-parameter Zhang neural networks for computing core and core-EP inverse. Neural Process. Lett. 51(2), 1299–1329 (2020)Zuo, K., Cvetković-Ilić, D., Cheng, Y.: Different characterizations of DMP-inverse of matrices. To appear in Linear and Multilinear Algebr