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Volume 15, Issue 2
The Relation Between a Tensor and Its Associated Semi-Symmetric Form

Hassan Bozorgmanesh, Masoud Hajarian & Anthony Theodore Chronopoulos

DOI: 10.4208/nmtma.OA-2021-0164

Numer. Math. Theor. Meth. Appl., 15 (2022), pp. 530-564.

Published online: 2022-03

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  • Abstract

It is known that every tensor has an associated semi-symmetric tensor. The purpose of this paper is to investigate the shared properties of a tensor and its semi-symmetric form. In particular, a corresponding semi-symmetric tensor has smaller Frobenius norm under some conditions and can be used to get smaller bounds for eigenvalues and solutions of dynamical systems and tensor complementarity problems. In addition, every tensor has the same eigenvalues as its corresponding semi-symmetric form, also a corresponding semi-symmetric tensor inherits properties like being circulant, Toeplitz, $Z$-tensor, $M$-tensor, $H$-tensor and some others. Also, there are a two-way connection for properties like being positive definite, $P$-tensor, semi-positive, primitive and several others.

  • Keywords

Tensor eigenvalue, semi-symmetric tensor, eigenvalue bound, logarithmic norm, tensor complementarity problem.

  • AMS Subject Headings

15A69, 15A03, 15A21, 15A72

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-15-530, author = {Bozorgmanesh , HassanHajarian , Masoud and Chronopoulos , Anthony Theodore}, title = {The Relation Between a Tensor and Its Associated Semi-Symmetric Form}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2022}, volume = {15}, number = {2}, pages = {530--564}, abstract = {

It is known that every tensor has an associated semi-symmetric tensor. The purpose of this paper is to investigate the shared properties of a tensor and its semi-symmetric form. In particular, a corresponding semi-symmetric tensor has smaller Frobenius norm under some conditions and can be used to get smaller bounds for eigenvalues and solutions of dynamical systems and tensor complementarity problems. In addition, every tensor has the same eigenvalues as its corresponding semi-symmetric form, also a corresponding semi-symmetric tensor inherits properties like being circulant, Toeplitz, $Z$-tensor, $M$-tensor, $H$-tensor and some others. Also, there are a two-way connection for properties like being positive definite, $P$-tensor, semi-positive, primitive and several others.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2021-0164}, url = {http://global-sci.org/intro/article_detail/nmtma/20363.html} }
TY - JOUR T1 - The Relation Between a Tensor and Its Associated Semi-Symmetric Form AU - Bozorgmanesh , Hassan AU - Hajarian , Masoud AU - Chronopoulos , Anthony Theodore JO - Numerical Mathematics: Theory, Methods and Applications VL - 2 SP - 530 EP - 564 PY - 2022 DA - 2022/03 SN - 15 DO - http://doi.org/10.4208/nmtma.OA-2021-0164 UR - https://global-sci.org/intro/article_detail/nmtma/20363.html KW - Tensor eigenvalue, semi-symmetric tensor, eigenvalue bound, logarithmic norm, tensor complementarity problem. AB -

It is known that every tensor has an associated semi-symmetric tensor. The purpose of this paper is to investigate the shared properties of a tensor and its semi-symmetric form. In particular, a corresponding semi-symmetric tensor has smaller Frobenius norm under some conditions and can be used to get smaller bounds for eigenvalues and solutions of dynamical systems and tensor complementarity problems. In addition, every tensor has the same eigenvalues as its corresponding semi-symmetric form, also a corresponding semi-symmetric tensor inherits properties like being circulant, Toeplitz, $Z$-tensor, $M$-tensor, $H$-tensor and some others. Also, there are a two-way connection for properties like being positive definite, $P$-tensor, semi-positive, primitive and several others.

Bozorgmanesh , HassanHajarian , Masoud and Chronopoulos , Anthony Theodore. (2022). The Relation Between a Tensor and Its Associated Semi-Symmetric Form. Numerical Mathematics: Theory, Methods and Applications. 15 (2). 530-564. doi:10.4208/nmtma.OA-2021-0164
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