Frontiers of Mathematics in China >

2020 , Vol. 15 >Issue 6: 1175 - 1188

DOI: https://doi.org/10.1007/s11464-020-0884-3

RESEARCH ARTICLE

On intervals and sets of hypermatrices (tensors)

  • Saeed RAHMATI ,
  • Mohamed A. TAWHID
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  • Department of Mathematics and Statistics, Faculty of Science, Thompson Rivers University, Kamloops, BC V2C 0C8, Canada

Received date: 15 Jul 2020

Accepted date: 07 Dec 2020

Published date: 15 Dec 2020

Copyright

2020 Higher Education Press
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Abstract

Interval hypermatrices (tensors) are introduced and interval α-hypermatrices are uniformly characterized using a finite set of 'extreme' hypermatrices, where α can be strong P, semi-positive, or positive definite, among many others. It is shown that a symmetric interval is an interval (strictly) copositive-hypermatrix if and only if it is an interval (E) E0-hypermatrix. It is also shown that an even-order, symmetric interval is an interval positive (semi-) definite-hypermatrix if and only if it is an interval P (P0)-hypermatrix. Interval hypermatrices are generalized to sets of hyper-matrices, several slice-properties of a set of hypermatrices are introduced and sets of hypermatrices with various slice-properties are uniformly characterized. As a consequence, several slice-properties of a compact, convex set of hyper-matrices are characterized by its extreme points.

Key words： Tensor; hypermatrix; interval hypermatrix; hypermatrix set; slice-P-property

Cite this article

Saeed RAHMATI , Mohamed A. TAWHID . On intervals and sets of hypermatrices (tensors)[J]. Frontiers of Mathematics in China, 2020 , 15(6) : 1175 -1188 . DOI: 10.1007/s11464-020-0884-3

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