报告主题：Third Order Tensors and Hypermatrices
报告人：祁立群 教授 (香港理工大学)
报告时间：2017年4月18日（周二）18:00
报告地点：校本部G507
邀请人：王卿文
主办部门：8455新葡萄场网站数学系
报告摘要：Third order tensors have wide applications in mechanics, physics and engineering. The most famous and useful third order tensor is the piezoelectric tensor, which plays a key role in the piezoelectric effect, first discovered by Curie brothers. On the other hand, the Levi-Civita tensor is famous in tensor calculus. In this talk, we study third order tensors and (third order) hypermatrices systematically, by regarding a third order tensor as a linear operator which transforms a second order tensor into a first order tensor, or a first order tensor into a second order tensor. For a third order tensor, we define its transpose, kernel tensor and L-inverse. The transpose of a third order tensor is uniquely defined. In particular, the transpose of the piezoelectric tensor is the inverse piezoelectric tensor (the electrostriction tensor).
The kernel tensor of a third order tensor is a second order positive semi-definite symmetric tensor, which is the product of that third order tensor and its transpose. We define non-singularity for a third order tensor. A third order tensor has an L-inverse if and only if it is nonsingular. Here, ``L'' is named after Levi-Civita. We also define L-eigenvalues, singular values, C-eigenvalues and Z-eigenvalues for a third order tensor. They are all invariants of that third order tensor. A third order tensor is nonsingular if and only if all of its L-eigenvalues are positive. Physical meanings of these new concepts are discussed. We show that the Levi-Civita tensor is nonsingular, its L-inverse is a half of itself, and its three L-eigenvalues are all the square root of two. We also introduce third order orthogonal tensors. Third order orthogonal tensors are nonsingular. Their L-inverses are their transposes.
欢迎教师、学生参加 ！
