Partial orders based on the CS decomposition

[EN] A new decomposition for square matrices is constructed by using two known matrix decompositions. A new characterization of the core-EP order is obtained by using this new matrix decomposition. We also use this matrix decomposition to investigate the minus, star, sharp and core partial orders in the setting of complex matrices.The present research is supported by the National Natural Science Foundation of China (No. 11771076). The first author is supported by the National Science Foundation of Jiangsu, Province of China (No. BK20191047) and the National Science Foundation of Jiangsu Education Committee (No. 19KJB110005).Xu, SZ.; Chen, JL.; Benítez López, J. (2021). Partial orders based on the CS decomposition. Ukrainian Mathematical Journal. 72(8):1294-1313. https://doi.org/10.1007/s11253-020-01851-5S12941313728J. Benítez, “A new decomposition for square matrices,” Electron. J. Linear Algebra, 20, 207–225 (2010).J. Benítez and X. J. Liu, “A short proof of a matrix decomposition with applications,” Linear Algebra Appl., 438, 1398–1414 (2013).J. K. Baksalary and S. K. Mitra, “Left-star and right-star partial orderings,” Linear Algebra Appl., 149, 73–89 (1991).O. M. Baksalary and G. Trenkler, “Core inverse of matrices,” Linear Multilinear Algebra, 58, No. 6, 681–697 (2010).R. E. Cline and R. E. Funderlic, “The rank of a difference of matrices and associated generalized inverses,” Linear Algebra Appl., 24, 185–215 (1979).H. B. Chen and Y. J. Wang, “A family of higher-order convergent iterative methods for computing the Moore–Penrose inverse,” Appl. Math. Comput., 218, 4012–4016 (2011).M. P. Drazin, “Natural structures on semigroups with involution,” Bull. Amer. Math. Soc. (N.S.), 84, No. 1, 139–141 (1978).R. E. Hartwig, “How to order regular elements?,” Math. Jap., 25, 1–13 (1980).R. E. Hartwig and K. Spindelböck, “Matrices for which A* and A† commute,” Linear Multilinear Algebra, 14, 241–256 (1984).R. E. Hartwig and J. Shoaf, “Group inverses and Drazin inverses of bidiagonal and triangular Toeplitz matrices,” J. Austral. Math. Soc., Ser. A, 24, 10–34 (1977).L. Lebtahi, P. Patrício, and N. Thome, “The diamond partial order in rings,” Linear Multilinear Algebra, 62, No. 3, 386–395 (2013).S. K. Mitra, “On group inverses and the sharp order,” Linear Algebra Appl., 92, 17–37 (1987).S. B. Malik, “Some more properties of core partial order,” Appl. Math. Comput., 221, 192–201 (2013).K. M. Prasad and K. S. Mohana, “Core-EP inverse,” Linear Multilinear Algebra, 62, No. 6, 792–802 (2014).G. Marsaglia and G. P. H. Styan, “Equalities and inequalities for ranks of matrices,” Linear Multilinear Algebra, 2, 269–292 (1974).D. S. Rakić and D. S. Djordjević, “Star, sharp, core, and dual core partial order in rings with involution,” Appl. Math. Comput., 259, 800–818 (2015).H. X. Wang, “Core-EP decomposition and its applications,” Linear Algebra Appl., 508, 289–300 (2016).H. K. Wimmer, “Canonical angles of unitary spaces and perturbations of direct complements,” Linear Algebra Appl., 287, 373–379 (1999).S. Z. Xu, J. L. Chen, and X. X. Zhang, “New characterizations for core inverses in rings with involution,” Front. Math. China, 12, No. 1, 231–246 (2017).X. X. Zhang, S. Z. Xu, and J. L. Chen, “Core partial order in rings with involution,” Filomat, 31, No. 18, 5695–5701 (2017)