Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors

By Haibin Chen, Liqun Qi. 2014

70 citations recorded by the source corpus.

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. Furthermore, we prove that the Hadamard product of two positive semi-definite (positive definite respectively) symmetric Cauchy tensors is a positive semi-definite (positive definite respectively) tensor, which can be generalized to the Hadamard product of finitely many positive semi-definite (positive definite respectively) symmetric Cauchy tensors. 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.

Subfields: Computational Mathematics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/997c578e53e6d0c36b05f2178fe540e96252b404

How researchers see this paper

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In Computational Mathematics

1. Yiju Wang
2. L. Caccetta
3. Guanglu Zhou
4. Liqun Qi

In Radiology, Nuclear Medicine and Imaging

1. Yiju Wang
2. L. Caccetta
3. Guanglu Zhou

Citations in the corpus

• Nonsingular $H$-tensor and its criteria (2016, background): “Hence, tensor analysis and computing have received much attention of researchers in recent decade and the research branches include spectral hypergraph theory [13, 29], best low-rank approximation to tensors [15, 17, 37], positive definiteness identification of even order homogenous polynomials [12,”