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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. 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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, No. 8, pp. 1119–1133, August, 2020. Ukrainian DOI: 10.37863/umzh.v72i8.6025.
Rights and permissions
About this article
Cite this article
Xu, S.Z., Chen, J.L. & Benítez, J. Partial Orders Based on the CS Decomposition. Ukr Math J 72, 1294–1313 (2021). https://doi.org/10.1007/s11253-020-01851-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-020-01851-5