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

doi:10.1007/s11464-017-0628-1

Front. Math. China

2017, Vol. 12

Issue (6) : 1319-1337 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 JI¹, Yimin WEI²(

)

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

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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.

Keywords Fundamental theorem weighted Moore-Penrose inverse multilinear system null space and range tensor equation

Corresponding Author(s): Yimin WEI

Issue Date: 27 November 2017

Cite this article:

Jun JI,Yimin WEI. Weighted Moore-Penrose inverses and fundamental theorem of even-order tensors with Einstein product[J]. Front. Math. China, 2017, 12(6): 1319-1337.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-017-0628-1
https://academic.hep.com.cn/fmc/EN/Y2017/V12/I6/1319

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