Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/28880
Title: Positive definiteness and semi-definiteness of even order symmetric cauchy tensors
Authors: Chen, H
Qi, L 
Keywords: Cauchy tensor
Eigenvalue
Generating vector
Positive definiteness
Positive semi-definiteness
Issue Date: 2015
Publisher: American Institute of Mathematical Sciences
Source: Journal of industrial and management optimization, 2015, v. 11, no. 4, p. 1263-1274 How to cite?
Journal: Journal of industrial and management optimization 
Abstract: Motivated by symmetric Cauchy matrices, we define symmetric Cauchy tensors and their generating vectors in this paper. Hilbert tensors are symmetric Cauchy tensors. An even order symmetric Cauchy tensor is positive semi-definite if and only if its generating vector is positive. An even order symmetric Cauchy tensor is positive definite if and only if its generating vector has positive and mutually distinct entries. This extends Fiedler's result for symmetric Cauchy matrices to symmetric Cauchy tensors. Then, it is proven that the positive semi-definiteness character of an even order symmetric Cauchy tensor can be equivalently checked by the monotone increasing property of a homogeneous polynomial related to the Cauchy tensor. The homogeneous polynomial is strictly monotone increasing in the nonnegative orthant of the Euclidean space when the even order symmetric Cauchy tensor is positive definite. At last, bounds of the largest H-eigenvalue of a positive semi-definite symmetric Cauchy tensor are given and several spectral properties on Z-eigenvalues of odd order symmetric Cauchy tensors are shown. Further questions on Cauchy tensors are raised.
URI: http://hdl.handle.net/10397/28880
ISSN: 1547-5816
EISSN: 1553-166X
DOI: 10.3934/jimo.2015.11.1263
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