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One-sided weighted outer inverses of tensors. (English) Zbl 1458.15010

The authors introduce one-sided \((M,N)\)-weighted \((B,C)\)-inverse of a tensor as wider classes of one-sided inverses. They give some characterizations for the existence of these new inverses. In addition, they present the sets of all one-sided weighted inverses of a given tensor. They then generalize the notation of \((B,C)\)-inverse of a complex tensor. In addition, they introduce a left and right \((B,C)\)-inverse of a tensor and \((B,C)\)-inverse of a tensor. When \(B=C\), they give the notations of a left and right invertible tensor along a given tensor \(B\). Moreover, they introduce an \((M,N)\)-weighted \((B,C)\)-outer inverse and W-weighted \((B,C)\)-outer inverse of tensors. Further, they obtain some characterizations of this tensor inverse. They also give some algorithms for calculating the introduced tensor outer inverses. Finally, they study a color image deblurring problem and find solutions to the Poisson problem in a tensor-structure domain.

MSC:

15A09 Theory of matrix inversion and generalized inverses
15A24 Matrix equations and identities
15A69 Multilinear algebra, tensor calculus
53A45 Differential geometric aspects in vector and tensor analysis
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[1] Ben-Israel, A.; Grevile, T. N.E., Generalized Inverses, Theory and Applications (2003), Canadian Mathematical Society, Springer: Canadian Mathematical Society, Springer New York, Beflin, Heidelberg, Hong Kong, London, Milan, Paris, Tokyo · Zbl 1026.15004
[2] Wang, G.; Wei, Y.; Qiao, S., (Generalized Inverses: Theory and Computations. Generalized Inverses: Theory and Computations, Developments in Mathematics, vol. 53 (2018), Springer, Science Press: Springer, Science Press Singapore, Beijing) · Zbl 1395.15002
[3] Katsikis, V. N.; Pappas, D.; Petralias, A., An improved method for the computation of the Moore-Penrose inverse matrix, Appl. Math. Comput., 217, 9828-9834 (2011) · Zbl 1220.65049
[4] Stanimirović, P. S.; Pappas, D.; Katsikis, V. N.; Stanimirović, I. P., Full-rank representations of outer inverses based on the QR decomposition, Appl. Math. Comput., 218, 10321-10333 (2012) · Zbl 1246.65067
[5] Stanimirović, P. S.; Tasić, M. B., Computing generalized inverses using LU factorization of matrix product, Int. J. Comput. Math., 85, 1865-1878 (2008) · Zbl 1158.65029
[6] Guo, W. B.; Huang, T. Z., Method of elementary transformation to compute Moore-Penrose inverse, Appl. Math. Comput., 216, 1614-1617 (2010) · Zbl 1200.65027
[7] Stanimirović, P. S.; Petković, M. D., Gauss-Jordan elimination method for computing outer inverses, Appl. Math. Comput., 219, 4667-4679 (2013) · Zbl 06447273
[8] Djordjević, D. S.; Stanimirović, P. S.; Wei, Y., The representation and approximation of outer generalized inverses, Acta Math. Hungar., 104, 1-26 (2004) · Zbl 1071.65075
[9] Li, W. G.; Li, Z., A family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non-square matrix, Appl. Math. Comput., 215, 3433-3442 (2010) · Zbl 1185.65057
[10] Petković, M. D.; Stanimirović, P. S., Two improvements of the iterative method for computing Moore-Penrose inverse based on Penrose equations, J. Comput. Appl. Math., 267, 61-71 (2014) · Zbl 1293.65048
[11] Stanimirović, P. S.; Cvetković-Ilić, D. S., Successive matrix squaring algorithm for computing outer inverses, Appl. Math. Comput., 203, 19-29 (2008) · Zbl 1158.65028
[12] Wei, Y.; Wang, G. R., Approximate methods for the generalized inverse \(A_{T , S}^{( 2 )}\), J. Fudan Univ. (Nat. Sci.), 38, 234-249 (1999) · Zbl 0961.65037
[13] Wei, Y.; Wu, H., The representation and approximation for the generalized inverse \(A_{T , S}^{( 2 )}\), Appl. Math. Comput., 135, 263-276 (2003) · Zbl 1027.65048
[14] Liu, X.; Yu, Y.; Zhong, J.; Wei, Y., Integral and limit representations of the outer inverse in Banach space, Linear Multilinear Algebra, 60, 333-347 (2012) · Zbl 1239.47002
[15] Stanimirović, P. S., Limit representations of generalized inverses and related methods, Appl. Math. Comput., 103, 51-68 (1999) · Zbl 0927.65060
[16] Najafi, H. S.; Solary, M. S., Computational algorithms for computing the inverse of a square matrix, quasi-inverse of a non-square matrix and block matrices, Appl. Math. Comput., 183, 539-550 (2006) · Zbl 1104.65309
[17] Xia, Y.; Zhang, S.; Stanimirović, P. S., Neural network for computing pseudoinverses and outer inverses of complex-valued matrices, Appl. Math. Comput., 273, 1107-1121 (2016) · Zbl 1410.65084
[18] Miao, Y.; Qi, L.; Wei, Y., Generalized tensor function via the tensor singular value decomposition based on the T-product, Linear Algebra Appl., 590, 1, 258-303 (2020) · Zbl 1437.15034
[19] Miao, Y.; Qi, L.; Wei, Y., T-Jordan canonical form and T-Drazin inverse based on the T-product, Commun. Appl. Math. Comput. (2020)
[20] Cao, C. G.; Zhang, X., The generalized inverse \(A_{T , \ast}^{( 2 )}\) and its applications, J. Appl. Math. Comput., 11, 155-164 (2003) · Zbl 1027.15005
[21] Mary, X., On generalized inverses and Green’s relations, Linear Algebra Appl., 434, 8, 1836-1844 (2011) · Zbl 1219.15007
[22] Drazin, M. P., A class of outer generalized inverses, Linear Algebra Appl., 436, 1909-1923 (2012) · Zbl 1254.15005
[23] Benítez, J.; Boasso, E., The inverse along an element in rings with an involution, Banach algebras and \(C^\ast \)-algebras, Linear Multilinear Algebra, 65, 2, 284-299 (2017) · Zbl 1361.15004
[24] Chen, J. L.; Ke, Y. Y.; Mosić, D., The reverse order law of the \(( b , c )\)-inverse in semigroups, Acta Math. Hungar., 151, 1, 181-198 (2017) · Zbl 1399.20065
[25] Mosić, D.; Zou, H.; Chen, J. L., On the \(( b , c )\)-inverse in rings, Filomat, 32, 4, 1221-1231 (2018) · Zbl 1499.16097
[26] Wang, L.; Castro-González, N.; Chen, J. L., Characterizations of outer generalized inverses, Canad. Math. Bull., 60, 861-871 (2017) · Zbl 1383.15006
[27] Zhu, H. H.; Chen, J. L.; Patrí cio, P., Further results on the inverse along an element in semigroups and rings, Linear Multilinear Algebra, 64, 393-403 (2016) · Zbl 1338.15014
[28] Drazin, M. P., Left and right generalized inverses, Linear Algebra Appl., 510, 64-78 (2016) · Zbl 1355.15003
[29] Benítez, J.; Boasso, E.; Jin, H., On one-sided \(( B , C )\)-inverses of arbitrary matrices, Electron. J. Linear Algebra, 32, 391-422 (2017) · Zbl 1386.15016
[30] Ding, W.; Wei, Y., Theory and Computation of Tensors (2016), Elsevier, Academic Press: Elsevier, Academic Press Amsterdam, Boston, Heidelberg, London, New York, Oxford, Paris, San Diego, San Francisco, Singapore, Sydney, Tokyo · Zbl 1380.15004
[31] Einstein, A., The foundation of the general theory of relativity, (Kox, A.; Klein, M.; Schulmann, R., The Collected Papers of Albert Einstein, Vol. 6 (2007), Princeton University Press: Princeton University Press Princeton (NJ)), 146-200
[32] Ji, J.; Wei, Y., Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product, Front. Math. China, 12, 1319-1337 (2017) · Zbl 1393.15007
[33] 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 (2020) · Zbl 1458.15011
[34] Brazell, M.; Li, N.; Navasca, C.; Tamon, C., Solving multilinear systems via tensor inversion, SIAM J. Matrix Anal. Appl., 34, 542-570 (2013) · Zbl 1273.15028
[35] Liang, M.; Zheng, B.; Zhao, R., Tensor inversion and its application to the tensor equations with Einstein product, Linear Multilinear Algebra, 67, 843-870 (2019) · Zbl 1411.15017
[36] Behera, R.; Mishra, D., Further results on generalized inverses of tensors via the Einstein product, Linear Multilinear Algebra, 65, 1662-1682 (2017) · Zbl 1370.15025
[37] Sun, L.; Zheng, B.; Bu, C.; Wei, Y., Moore-Penrose inverse of tensors via Einstein product, Linear Multilinear Algebra, 64, 686-698 (2016) · Zbl 1341.15019
[38] Panigrahy, K.; Mishra, D., An extension of the Moore-Penrose inverse of a tensor via the Einstein product, Linear Multilinear Algebra (2020)
[39] Ji, J.; Wei, Y., The Drazin inverse of an even-order tensor and its application to singular tensor equations, Comput. Math. Appl., 75, 3402-3413 (2018) · Zbl 1408.15002
[40] Sahoo, J. K.; Behera, R.; Stanimirović, P. S.; Katsikis, V., Computation of outer inverses of tensors using the QR decomposition, Comput. Appl. Math., 39 (2020) · Zbl 1463.15009
[41] Mosić, D.; Djordjević, D. S., Weighted outer inverse, Monatsh. Math., 188, 2, 297-307 (2019) · Zbl 07004942
[42] Behera, R.; Maji, S.; Mohapatra, R. N., Weighted Moore-Penrose inverses of arbitrary-order tensors, Comp. Appl. Math., 39 (2020) · Zbl 1476.65062
[43] Behera, R.; Nandi, A. K.; Sahoo, J. K., Further results on the Drazin inverse of even order tensors, Numer. Linear Algebra Appl., 27 (2020) · Zbl 1474.15063
[44] Rakočević, V.; Wei, Y., The representation and approximation of the W-weighted Drazin inverse of linear operators in Hilbert space, Appl. Math. Comput., 141, 455-470 (2003) · Zbl 1044.47003
[45] Ke, Y.; Chen, J.; Stanimirović, P. S.; Ćirić, M., Characterizations and representations of outer inverse for matrices over a ring, Linear Multilnear Algebra (2019)
[46] Beik, F. P.A.; Jbilou, K.; Najafi-Kalyani, M.; Reichel, L., Golub-Kahan bidiagonalization for ill-conditioned tensor equations with applications, Numer. Algorithms (2020) · Zbl 1483.65069
[47] Najafi-Kalyani, M.; Beik, F. P.A.; Jbilou, K., On global iterative schemes based on Hessenberg process for (ill-posed) Sylvester tensor equations, J. Comput. Appl. Math. (2019)
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