期刊
LINEAR ALGEBRA AND ITS APPLICATIONS
卷 457, 期 -, 页码 303-312出版社
ELSEVIER SCIENCE INC
DOI: 10.1016/j.laa.2014.05.026
关键词
Positive definiteness; B tensor; B-0 tensor
资金
- Hong Kong Research Grant Council [PolyU 502510, 502111, 501212, 501913]
- National Natural Science Foundation of China [11171094, 11271112]
It is easily checkable if a given tensor is a B tensor, or a B-0 tensor or not. In this paper, we show that a symmetric B tensor can always be decomposed to the sum of a strictly diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors, and a symmetric B-0 tensor can always be decomposed to the sum of a diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors. When the order is even, this implies that the corresponding B tensor is positive definite, and the corresponding B-0 tensor is positive semi-definite. This gives a checkable sufficient condition for positive definite and semi-definite tensors. This approach is different from the approach in the literature for proving a symmetric B matrix is positive definite, as that matrix approach cannot be extended to the tensor case. (C) 2014 Elsevier Inc. All rights reserved.
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