Further investigation of positive semi-definiteness of fourth-order Cauchy and Hilbert tensors. (English) Zbl 1526.15027
Summary: In this paper, we further investigate the positive (semi-)definiteness of even order tensors. For the fourth-order tensor, we discuss the relationship between positive (semi-)definiteness and extensively positive (semi-)definiteness. Moreover, we give several necessary and sufficient conditions for fourth-order generalized Cauchy tensors to be extensively positive (semi-)definite. Furthermore, it is proved that fourth-order generalized Cauchy tensors are extensively positive semi-definite if and only if the associated homogeneous polynomial is monotonically increasing. At last, we prove that infinite dimensional generalized Hilbert tensors are also positive definite under other conditions, and present the upper bounds on the norms of positively homogeneous operators related to the infinite dimensional generalized Hilbert tensor.
MSC:
| 15A69 | Multilinear algebra, tensor calculus |
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