Abstract
We define the {i}-inverse (i = 1, 2, 5) and group inverse of tensors based on a general product of tensors. We explore properties of the generalized inverses of tensors on solving tensor equations and computing formulas of block tensors. We use the {1}-inverse of tensors to give the solutions of a multilinear system represented by tensors. The representations for the {1}-inverse and group inverse of some block tensors are established.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Behera R, Mishra D. Further results on generalized inverses of tensors via Einstein product. Linear Multilinear Algebra, 2017, 65(8): 1662–1682
Beltrán C, Shub M. On the geometry and topology of the solution variety for polynomial system solving. Found Comput Math, 2017, 12(6): 719–763
Ben-Isral A, Greville T N E. Generalized Inverse: Theory and Applications. New York: Wiley-Interscience, 1974
Brazell M, Li N, Navasca C, Tamon C. Solving multilinear systems via tensor inversion. SIAM J Matrix Anal Appl, 2013, 34(2): 542–570
Bu C, Wang W, Sun L, Zhou J. Minimum (maximum) rank of sign pattern tensors and sign nonsingular tensors. Linear Algebra Appl, 2015, 483: 101–114
Bu C, Wei Y P, Sun L, Zhou J. Brualdi-type eigenvalue inclusion sets of tensors. Linear Algebra Appl, 2015, 480: 168–175
Bu C, Zhang X, Zhou J, Wang W, Wei Y. The inverse, rank and product of tensors. Linear Algebra Appl, 2014, 446: 269–280
Campbell S L. The Drazin inverse and systems of second order linear differential equations. Linear Multilinear Algebra, 1983, 14: 195–198
Campbell S L, Meyer C D. Generalized Inverses of Linear Transformations. London: Pitman, 1979
Chang K C, Pearson K, Zhang T. Primitivity, the convergence of the NQZ method, and the largest eigenvalue for nonnegative tensors. SIAM J Matrix Anal Appl, 2011, 32(3): 806–819
Che M, Bu C, Qi L, Wei Y. Nonnegative tensors revisited: plane stochastic tensors. Linear Multilinear Algebra, https://doi.org/10.1080/03081087.2018.1453469
Chen J, Saad Y. On the tensor SVD and the optimal low rank orthogonal approximation of tensors. SIAM J Matrix Anal Appl, 2009, 30(4): 1709–1734
Chen Y, Qi L, Zhang X. The fielder vector of a Laplacian tensor for hypergraph partitioning. SIAM J Sci Comput, 2017, 39(6): A2508–A2537
Cooper J, Dutle A. Spectra of uniform hypergraphs. Linear Algebra Appl, 2012, 436: 3268–3292
Ding W, Wei Y. Solving multi-linear systems with M-tensors. J Sci Comput, 2016, 68: 689–715
Fan Z, Deng C, Li H, Bu C. Tensor representations of quivers and hypermatrices. Preprint
Hartwig R E, Shoaf J M. Group inverse and Drazin inverse of bidiagonal and triangular Toeplitz matrices. Aust J Math, 1977, 24(A): 10–34
Hu S, Huang Z, Ling C, Qi L. On determinants and eigenvalue theory of tensors. J Symbolic Comput, 2013, 50: 508–531
Huang S, Zhao G, Chen M. Tensor extreme learning design via generalized Moore-Penrose inverse and triangular type-2 fuzzy sets. Neural Comput Appl, https://doi.org/10.1007/s00521-018-3385-5
Hunter J J. Generalized inverses of Markovian kernels in terms of properties of the Markov chain. Linear Algebra Appl, 2014, 447: 38–55
Ji J, Wei Y. Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product. Front Math China, 2017, 12(6): 1319–1337
Ji J, Wei Y. The Drazin inverse of an even-order tensor and its application to singular tensor equations. Comput Math Appl, 2018, 75: 3402–3413
Kirkland S J, Neumann M, Shader B L. On a bound on algebraic connectivity: the case of equality. Czechoslovak Math J, 1998, 48: 65–76
Kroonenberg P M. Applied Multiway Data Analysis. New York: Wiley-Interscience, 2008
Liu W, Li W. On the inverse of a tensor. Linear Algebra Appl, 2016, 495: 199–205
Lu H, Plataniotis K N, Venetsanopoulos A. Multilinear Subspace Learning: Dimensionality Reduction of Multidimensional Data. New York: CRC Press, 2013
Qi L. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324
Qi L, Luo Z. Tensor Analysis: Spectral Theory and Special Tensors. Philadelphia: SIAM, 2017
Shao J. A general product of tensors with applications. Linear Algebra Appl, 2013, 439: 2350–2366
Shao J, You L. On some properties of three different types of triangular blocked tensors. Linear Algebra Appl, 2016, 511: 110–140
Sidiropoulos N D, Lathauwer L D, Fu X, Huang K, Papalexakis E E, Faloutsos C. Tensor decomposition for signal processing and machine learning. IEEE Trans Signal Process, 2017, 65(13): 3551–3582
Sun L, Wang W, Zhou J, Bu C. Some results on resistance distances and resistance matrices. Linear Multilinear Algebra, 2015, 63(3): 523–533
Sun L, Zheng B, Bu C, Wei Y. Moore-Penrose inverse of tensors via Einstein product. Linear Multilinear Algebra, 2016, 64(4): 686–698
Wei Y. Generalized inverses of matrices. In: Hogben L, ed. Handbook of Linear Algebra. 2nd ed. Boca Raton: Chapman and Hall/CRC, 2014, Chapter 27
Acknowledgements
The authors are very grateful to the referees for their valuable suggestions, which have considerably improved the paper. Yimin Wei was supported by the International Cooperation Project of Shanghai Municipal Science and Technology Commission (Grant No. 16510711200) and the National Natural Science Foundation of China (Grant No. 11771099); Changjiang Bu was supported by the National Natural Science Foundation of China (Grant No. 11371109).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sun, L., Zheng, B., Wei, Y. et al. Generalized inverses of tensors via a general product of tensors. Front. Math. China 13, 893–911 (2018). https://doi.org/10.1007/s11464-018-0695-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-018-0695-y