Frontiers of Mathematics in China >

2017 , Vol. 12 >Issue 6: 1319 - 1337

DOI: https://doi.org/10.1007/s11464-017-0628-1

RESEARCH ARTICLE

Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product

  • Jun JI 1 ,
  • Yimin WEI , 2
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  • 1. Department of Mathematics, Kennesaw State University, Marietta, GA 30060, USA
  • 2. School of Mathematical Sciences and Shanghai Key Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai 200433, China

Received date: 15 Sep 2016

Accepted date: 06 Jan 2017

Published date: 27 Nov 2017

Copyright

2017 Higher Education Press and Springer-Verlag GmbH Germany
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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) leastsquares (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.

Key words： Fundamental theorem; weighted Moore-Penrose inverse; multilinear system; null space and range; tensor equation

Cite this article

Jun JI , Yimin WEI . Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product[J]. Frontiers of Mathematics in China, 2017 , 12(6) : 1319 -1337 . DOI: 10.1007/s11464-017-0628-1

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
Ben-IsraelA, GrevilleT N E. Generalized Inverse: Theory and Applications.New York: John Wiley, 2003

2
BrazellM, LiN, NavascaC, TamonC. Solving multilinear systems via tensor inversion.SIAM J Matrix Anal Appl, 2013, 34: 542–570

DOI

3
BurdickD, McGownL, MillicanD, TuX. Resolution of multicomponent fluorescent mixtures by analysis of the excitation-emission-frequency array.J Chemometrics, 1990, 4: 15–28

DOI

4
ComonP. 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

5
CooperJ, DutleA. Spectra of uniform hypergraphs.Linear Algebra Appl, 2012, 436: 3268–3292

DOI

6
EinsteinA. 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

7
EldénL. Matrix Methods in Data Mining and Pattern Recognition.Philadelphia: SIAM, 2007

DOI

8
HuS, QiL. Algebraic connectivity of an even uniform hypergraph.J Comb Optim, 2012, 24: 564–579

DOI

9
KoldaT, BaderB. Tensor decompositions and applications.SIAM Review, 2009, 51: 455–500

DOI

10
LuoZ, QiL, YeY. Linear operators and positive semidefiniteness of symmetric tensors spaces.Sci China Math, 2015, 58: 197–212

DOI

11
SmildeA, BroR, GeladiP. Multi-Way Analysis: Applications in the Chemical Sciences.West Sussex: Wiley, 2004

DOI

12
SunL, ZhengB, BuC, WeiY. Moore-Penrose inverse of tensors via Einstein product.Linear Multilinear Algebra, 2016, 64: 686–698

DOI

13
VlasicD, BrandM, PfisterH, PopovicJ. Face transfer with multilinear models.ACM Trans Graphics, 2005, 24: 426–433

DOI

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