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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2016, Vol. 33 ›› Issue (3): 287-297.doi: 10.3969/j.issn.1005-3085.2016.03.007

• • 上一篇    下一篇

$\mathcal{H}$-张量的判定及其应用

王  峰,  孙德淑   

  1. 贵州民族大学理学院，贵阳 550025
  • 收稿日期:2015-03-25 接受日期:2015-09-10 出版日期:2016-06-15 发布日期:2016-08-15
  • 基金资助:
    国家自然科学基金 (11361074; 11501141)；贵州省科学技术基金 ([2015]2073)；贵州省教育厅自然科学基金 ([2015]420)；贵州民族大学科研基金 (15XRY004).

Criteria for $\mathcal{H}$-tensors and its Application

WANG Feng,   SUN De-shu   

  1. College of Science, Guizhou Minzu University, Guiyang 550025
  • Received:2015-03-25 Accepted:2015-09-10 Online:2016-06-15 Published:2016-08-15
  • Supported by:
    The National Natural Science Foundation of China (11361074; 11501141); the Science and Technology Programs of Guizhou Province ([2015]2073); the Foundation of the Education Department of Guizhou Province ([2015]420); the Research Foundation of Guizhou Minzu University (15XRY004).

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摘要： $\mathcal{H}$-张量在自动控制系统的稳定性、多项式全局优化、医疗影像降噪等问题中具有重要的应用价值．但在实际中要判定一给定张量为$\mathcal{H}$-张量比较困难．本文通过构造不同的正对角阵和运用不等式的放缩技巧，给出了$\mathcal{H}$-张量一组新的实用判定方法．作为应用，给出了偶数阶实对称张量正定性的新判定条件．相应数值示例说明了结果的有效性．

关键词: $\mathcal{H}$-张量, 实对称张量, 正定性, 不可约, 非零元素链

Abstract:

$\mathcal{H}$-tensors play an important role in the stability study of automatic control systems, polynomial optimization problems, medical image de-noising problems, and so on. But it is not easy to determine whether a given tensor is an $\mathcal{H}$-tensor or not in practice. In this paper, we give some practical criteria for $\mathcal{H}$-tensors by constructing different positive diagonal matrices and applying some techniques of inequalities. As an application, some sufficient conditions for the positive definiteness of an even-order real symmetric tensor are given. Advantages of obtained results are illustrated by numerical examples.

Key words: $\mathcal{H}$-tensors, real symmetric tensors, positive definiteness, irreducible, nonzero elements chain

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