Computing Tensor Generalized Bilateral Inverses

doi:10.1007/s42967-024-00373-2

Communications on Applied Mathematics and Computation ›› 2025, Vol. 7 ›› Issue (6): 2269-2288.doi: 10.1007/s42967-024-00373-2

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Computing Tensor Generalized Bilateral Inverses

Ratikanta Behera¹, Jajati Keshari Sahoo², Predrag S. Stanimirović^3,4, Alena Stupina⁴, Artem Stupin⁴

1. Department of Computational and Data Sciences, Indian Institute of Science, Bangalore, India;
2. Department of Mathematics, Birla Institute of Technology and Science Pilani, K. K. Birla Goa Campus, Goa, India;
3. Faculty of Sciences and Mathematics, University of Niš, Niš, Serbia;
4. Laboratory “Hybrid Methods of Modelling and Optimization in Complex Systems”, Siberian Federal University, Prosp. Svobodny 79, 660041, Krasnoyarsk, Russian Federation

Received:2023-08-31 Revised:2024-01-14 Published:2025-12-24
Contact: Predrag S. Stanimirović, E-mail:pecko@pmf.ni.ac.rs E-mail:pecko@pmf.ni.ac.rs
Supported by:
This research is supported by the Ministry of Science and Higher Education of the Russian Federation (Grant No. 075-15-2022-1121).

Abstract

Abstract: We introduce tensor generalized bilateral inverses (TGBIs) under the Einstein tensor product as an extension of generalized bilateral inverses (GBIs) in the matrix environment. Moreover, the TBGI class includes so far considered composite generalized inverses (CGIs) for matrices and tensors. Applications of TBGIs for solving multilinear systems are presented. The characterizations and representations of the TGBI were studied and verified using a specific algebraic approach. Further, a few characterizations of known CGIs (such as the CMP, DMP, MPD, MPCEP, and CEPMP) are derived. The main properties of the TGBIs were exploited and verified through numerical examples.

Key words: Generalized bilateral inverses (GBIs), Einstein product, Outer inverse, Poisson equations

CLC Number:

Ratikanta Behera, Jajati Keshari Sahoo, Predrag S. Stanimirović, Alena Stupina, Artem Stupin. Computing Tensor Generalized Bilateral Inverses[J]. Communications on Applied Mathematics and Computation, 2025, 7(6): 2269-2288.

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References

[1] Behera, R., Mishra, D.: Further results on generalized inverses of tensors via the Einstein product. Linear Multilinear Algebra 65(8), 1662–1682 (2017)
[2] Behera, R., Nandi, A.K., Sahoo, J.K.: Further results on the Drazin inverse of even-order tensors. Numer. Linear Algebra Appl. 27(5), e2317, 25 (2020)
[3] Brazell, M., Li, N., Navasca, C., Tamon, C.: Solving multilinear systems via tensor inversion. SIAM J. Matrix Anal. Appl. 34(2), 542–570 (2013)
[4] Chen, J., Mosić, D., Xu, S.: On a new generalized inverse for Hilbert space operators. Quaest. Math. 43(9), 1331–1348 (2020)
[5] Du, H.-M., Wang, B.-X., Ma, H.-F.: Perturbation theory for core and core-EP inverses of tensor via Einstein product. Filomat 33(16), 5207–5217 (2019)
[6] Einstein, A.: The foundation of the general theory of relativity. Ann. Phys. 49(7), 769–822 (1916)
[7] Ji, J., Wei, Y.: The Drazin inverse of an even-order tensor and its application to singular tensor equations. Comput. Math. Appl. 75(9), 3402–3413 (2018)
[8] Ji, J., Wei, Y.: The outer generalized inverse of an even-order tensor with the Einstein product through the matrix unfolding and tensor folding. Electron. J. Linear Algebra 36, 599–615 (2020)
[9] Kheirandish, E., Salemi, A.: Generalized bilateral inverses. J. Comput. Appl. Math. 428, 115137 (2023)
[10] Kheirandish, E., Salemi, A.: Generalized bilateral inverses of tensors via Einstein product with applications to singular tensor equations. Comput. Appl. Math. 42(8), 343 (2023)
[11] Lai, W.M., Rubin, D.H., Krempl, E., Rubin, D.: Introduction to Continuum Mechanics. Butterworth-Heinemann (2009)
[12] Ma, H., Li, N., Stanimirović, P.S., Katsikis, V.N.: Perturbation theory for Moore-Penrose inverse of tensor via Einstein product. Comput. Appl. Math. 38(3), 111, 24 (2019)
[13] Sahoo, J.K., Behera, R., Stanimirović, P.S., Katsikis, V.N., Ma, H.: Core and core-EP inverses of tensors. Comput. Appl. Math. 39(1), 9, 28 (2020)
[14] Stanimirović, P.S., Ćirić, M., Katsikis, V.N., Li, C., Ma, H.: Outer and (b, c) inverses of tensors. Linear Multilinear Algebra 68, 940–971 (2018)
[15] Sun, L., Zheng, B., Bu, C., Wei, Y.: Moore-Penrose inverse of tensors via Einstein product. Linear Multilinear Algebra 64(4), 686–698 (2016)
[16] Sun, L., Zheng, B., Wei, Y., Bu, C.: Generalized inverses of tensors via a general product of tensors. Front. Math. Chin. 13(4), 893–911 (2018)
[17] Wang, B., Du, H., Ma, H.: Perturbation bounds for DMP and CMP inverses of tensors via Einstein product. Comput. Appl. Math. 39(1), 28, 17 (2020)
[18] Wang, Y., Wei, Y.: Generalized eigenvalue for even order tensors via Einstein product and its applications in multilinear control systems. Comput. Appl. Math. 41(8), 419, 30 (2022)

Computing Tensor Generalized Bilateral Inverses

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