Outer and (b,c) inverses of tensors

ABSTRACT

Some basic properties of the range and null space of multidimensional arrays (tensors) with respect to Einstein tensor product are investigated. Also, an useful and effective definition of the tensor rank, termed as reshaping rank, is introduced. Further, computation of tensor outer with prescribed range and/or kernel of higher order tensors is considered. Starting from an algebraic approach, some new relationships between the problems of solving tensor equations and computation of tensor outer inverses with prescribed range and/or null pace are established. Conditions for the existence, representation and computation of the Moore-Penrose inverse, the weighted Moore-Penrose inverse, the Drazin inverse and the usual inverse of tensors are derived as corollaries. In this way, we derive extensions of known representations of various classes of matrix generalized inverses to multidimensional arrays. Effective algorithms for computing inner inverses of tensors as well as initiated algorithms for computing outer inverses of tensors are presented. Implementation of proposed algorithms in Matlab are developed and illustrative numerical examples are given. In addition, results related with the $(b,c)$-inverses on semigroups are examined in details in a specific semigroup of tensors with a binary associative operation defined as the Einstein tensor product.

COMMUNICATED BY:

• Chi-Kwong Li

Keywords:

• Multidimensional array
• generalized inverse
• tensor rank
• tensor equation
• Einstein product

AMS subject classifications:

• 15A09
• 15A69
• 53A45

Acknowledgments

This research is supported by the bilateral project between China and Serbia “The theory of tensors, operator matrices and applications (no. 4–5)”.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first and second author gratefully acknowledge support from the Ministry of Education and Science, Republic of Serbia, Grant No. 174013. The fourth author is supported by National Natural Science Foundation of China under grants 11601473. The fifth author is supported by National Natural Science Foundation of China under grants 11401143.