| 报告题目： | Hankel Tensors: Associated Hankel Matrices and Vandermonde Decomposition |
| 报 告 人： | 祁力群 教授 |
| 香港理工大学 | |
| 报告时间： | 10月30日上午10:00 |
| 报告地点： | 九龙湖数学系第一报告厅 |
| 相关介绍： | 报告人简介：祁力群教授于1968年在清华大学获得计算数学学士学位，并于1981年和1984年在Wisconsin-Madison大学获得计算科学硕士和博士学位。祁力群教授曾在清华大学，Wisconsin- Madison大学， New South Wales大学和香港城市大学任教，现在他于香港理工大学应用数学系担任首席教授（Chair Professor）。祁力群教授现已在各类国际期刊上发表了200多篇论文。他建立了半光滑牛顿法的超线性和二次收敛性理论。现在，祁力群教授的主要研究方向逐渐转移到张量的理论和应用上。祁力群教授的论文被世界范围的研究者广泛引用，他是世界上被引用次数最高的前300位数学家中的一位。 报告摘要是：Hankel tensors arise from applications such as signal processing. In this paper, we make an initial study on Hankel tensors. For each Hankel tensor, we associate it with a Hankel matrix and a higher order two-dimensional symmetric tensor, which we call the associated plane tensor. If the associated Hankel matrix is positive semi-definite, we call such a Hankel tensor a strong Hankel tensor. We show that an $m$ order $n$-dimensional tensor is a Hankel tensor if and only if it has a Vandermonde decomposition. We call a Hankel tensor a complete Hankel tensor if it has a Vandermonde decomposition with positive coefficients. We prove that if a Hankel tensor is copositive or an even order Hankel tensor is positive semi-definite, then the associated plane tensor is copositive or positive semi-definite, respectively. We show that even order strong and complete Hankel tensors are positive semi-definite, and the Hadamard product of two strong Hankel tensors is also a strong Hankel tensor. We show that all the H-eigenvalue of a complete Hankel tensors (maybe of odd order) are nonnegative. We give some upper bounds and lower bounds for the smallest and the largest Z-eigenvalues of a Hankel tensor, respectively. Further questions on Hankel tensors are raised. |
