Abstract
Specific definitions of the core and core-EP inverses of complex tensors are introduced. Some characterizations, representations and properties of the core and core-EP inverses are investigated. The results are verified using specific algebraic approach, based on proposed definitions and previously verified properties. The approach used here is new even in the matrix case.
Similar content being viewed by others
References
Baksalary OM, Trenkler G (2010) Core inverse of matrices. Linear Multilinear Algebra 58:681–697
Baksalary OM, Trenkler G (2014) On a generalized core inverse. Appl Math Comput 236:450–457
Behera R, Mishra D (2017) Further results on generalized inverses of tensors via the Einstein product. Linear Multilinear Algebra 65:1662–1682
Behera R, Nandi AK, Sahoo JK. Further results on the Drazin inverse of even-order tensors, Submitted, arXiv:1904.10783
Ben-Israel A, Greville TN (2003) Generalized inverses: theory and applications. Wiley, New York, 1974; 2nd edition, Springer
Brazell M, Li N, Navasca C, Tamon C (2013) Solving multilinear systems via tensor inversion. SIAM J Matrix Anal Appl 34:542–570
Chen JL, Zhuang GF, Wei YM (2009) The Drazin inverse of a sum of morphisms. Acta Math Sci Ser A (Chin Ed) 29(3):538–552
Cline RE (1968) Inverses of rank invariant powers of a matrix. SIAM J Numer Anal 5(1):182–197
Drazin MP (1958) Pseudo-inverses in associative rings and semigroups. Am Math Monthly 65:506–514
Einstein A (1916) The foundation of the general theory of relativity. Annalen der Physik 49:769–822
Ferreyra DE, Levis FE, Thome N (2018) Revisiting the core EP inverse and its extension to rectangular matrices. Quaest Math 41:265–281
Gao Y, Chen J (2018) Pseudo core inverses in rings with involution. Commun Algebra 46:38–50
Gao Y, Chen J, Patricio P (2018) Weighted core-ep inverse of a rectangular matrix, arXiv preprint arXiv:1803.10186
Gao YF, Chen JL, Patricio P (2018) Continuity of the core-EP inverse and its applications. arXiv:1805.08696
Ji J, Wei Y (2017) Weighted Moore–Penrose inverses and fundamental theorem of even-order tensors with Einstein product. Front Math China 12:1319–1337
Ji J, Wei Y (2018) The Drazin inverse of an even-order tensor and its application to singular tensor equations. Comput Math Appl 75:3402–3413
Kurata H (2018) Some theorems on the core inverse of matrices and the core partial ordering. Appl Math Comput 316:43–51
Lai W, Rubin D, Krempl E (2009) Introduction to continuum mechanics. Butterworth Heinemann, Oxford
Liang M, Zheng B, Zhao R (2019) Tensor inversion and its application to the tensor equations with Einstein product. Linear Multilinear Algebra 67:843–870
Ma H (2018) Optimal perturbation bounds for the core inverse. Appl Math Comput 336:176–181
Ma H (2019) A characterization and perturbation bounds for the weighted core-EP inverse. Quaestiones Mathematicae. https://doi.org/10.2989/16073606.2019.1584773
Ma H, Stanimirović PS (2019) Characterizations, approximation and perturbations of the core-EP inverse. Appl Math Comput 359:404–417
Qi L, Luo Z (2017) Tensor analysis: spectral theory and special tensors. Vol. 151. SIAM
Panigrahy K, Mishra D (2018) An extension of the Moore-Penrose inverse of a tensor via the Einstein product. arXiv:1806.03655v1, 10
Pappas D, Katsikis VN, Stanimirović PS (2018) The \(\lambda \)-Aluthge transform of EP matrices. Filomat 32:4403–4411
Prasad KM, Mohana KS (2014) Core-EP inverse. Linear Multilinear Algebra 62:792–802
Prasad KM, Raj MD (2018) Bordering method to compute core-EP inverse. Special Matrices 6:193–200
Prasad KM, Raj MD, Vinay M (2018) Iterative method to find core-EP inverse. Bull Kerala Math Assoc Spec Issue 16(1):139–152
Rakić DS, Dinčić NČ, Djordjević DS (2014) Group, Moore–Penrose, core and dual core inverse in rings with involution. Linear Algebra Appl 463:115–133
Rakić DS, Dinčić NČ, Djordjević DS (2014) Core inverse and core partial order of Hilbert space operators. Appl Math Comput 244:283–302
Stanimirovic P, Ciric M, Katsikis V, Li C, Ma H. Outer and (b,c) inverses of tensors. Linear Multilinear Algebra. https://doi.org/10.1080/03081087.2018.1521783
Sun L, Zheng B, Bu C, Wei Y (2016) Moore–Penrose inverse of tensors via Einstein product. Linear Multilinear Algebra 64:686–698
Tian Y, Wang H (2011) Characterizations of EP matrices and weighted-EP matrices. Linear Algebra Appl 434:1295–1318
Wang H, Liu X (2015) Characterizations of the core inverse and the core partial ordering. Linear Multilinear Algebra 63:1829–1836
Wei Y, Ding W (2016) Theory and computation of tensors. Elsevier, New York
Xu S, Chen JL, Zhang X (2017) New characterizations for core inverses in rings with involution. Front Math China 12:231–246
Zhou MM, Chen JL, Li TT, Wang DG (2018) Three limit representations of the core-EP inverse. Filomat 32:5887–5894
Acknowledgements
Ratikanta Behera acknowledges the support provided by Science and Engineering Research Board (SERB), Department of Science and Technology, India, under the Grant No. EEQ/2017/000747.
Predrag Stanimirović gratefully acknowledges support from the Ministry of Education, Science and Technological Development, Republic of Serbia, Grant No. 174013.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jinyun Yuan.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Sahoo, J.K., Behera, R., Stanimirović, P.S. et al. Core and core-EP inverses of tensors. Comp. Appl. Math. 39, 9 (2020). https://doi.org/10.1007/s40314-019-0983-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-019-0983-5