Abstract
We treat even-order tensors with Einstein product as linear operators from tensor space to tensor space, define the null spaces and the ranges of tensors, and study their relationship. We extend the fundamental theorem of linear algebra for matrix spaces to tensor spaces. Using the new relationship, we characterize the least-squares (ℳ) solutions to a multilinear system and establish the relationship between the minimum-norm (N) least-squares (ℳ) solution of a multilinear system and the weighted Moore-Penrose inverse of its coefficient tensor. We also investigate a class of even-order tensors induced by matrices and obtain some interesting properties.
Article PDF
Similar content being viewed by others
References
Ben-Israel A, Greville T N E. Generalized Inverse: Theory and Applications. New York: John Wiley, 2003
Brazell M, Li N, Navasca C, Tamon C. Solving multilinear systems via tensor inversion. SIAM J Matrix Anal Appl, 2013, 34: 542–570
Burdick D, McGown L, Millican D, Tu X. Resolution of multicomponent fluorescent mixtures by analysis of the excitation-emission-frequency array. J Chemometrics, 1990, 4: 15–28
Comon P. Tensor decompositions: State of the art and applications. In: McWhirter J G, Proudler I K, eds. Mathematics in Signal Processing, V. Oxford: Oxford Univ Press, 2001, 1–24
Cooper J, Dutle A. Spectra of uniform hypergraphs. Linear Algebra Appl, 2012, 436: 3268–3292
Einstein A. The foundation of the general theory of relativity. In: Kox A J, Klein M J, Schulmann R, eds. The Collected Papers of Albert Einstein. Princeton: Princeton Univ Press, 2007, 146–200
Eldén L. Matrix Methods in Data Mining and Pattern Recognition. Philadelphia: SIAM, 2007
Hu S, Qi L. Algebraic connectivity of an even uniform hypergraph. J Comb Optim, 2012, 24: 564–579
Kolda T, Bader B. Tensor decompositions and applications. SIAM Review, 2009, 51: 455–500
Luo Z, Qi L, Ye Y. Linear operators and positive semidefiniteness of symmetric tensors spaces. Sci China Math, 2015, 58: 197–212
Smilde A, Bro R, Geladi P. Multi-Way Analysis: Applications in the Chemical Sciences. West Sussex: Wiley, 2004
Sun L, Zheng B, Bu C, Wei Y. Moore-Penrose inverse of tensors via Einstein product. Linear Multilinear Algebra, 2016, 64: 686–698
Vlasic D, Brand M, Pfister H, Popovic J. Face transfer with multilinear models. ACM Trans Graphics, 2005, 24: 426–433
Acknowledgements
The research of Jun Ji was partly supported by the Kennesaw State University Tenured Faculty Professional Development Full Paid Leave Program in Fall 2015; Yimin Wei was supported by the International Cooperation Project of Shanghai Municipal Science and Technology Commission (Grant No. 16510711200).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s11464-017-0628-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11464-017-0628-1
Keywords
- Fundamental theorem
- weighted Moore-Penrose inverse
- multilinear system
- null space and range
- tensor equation