Liqun Qi’s scientific contributions

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 paper, 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 to a first order tensor, and a first order tensor to a second order tensor. We introduce the transpose, the kernel tensor and the inverse of a third order tensor. The transpose of a third order tensor is uniquely defined. 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 inverse if and only if it is nonsingular. We also define 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 eigenvalues are positive. Physical meanings of these new concepts are discussed. We show that the Levi-Civita tensor is nonsingular, its inverse is a half of itself, and its three eigenvalues are all the square root of two. We also introduce third order orthogonal tensors. Third order orthogonal tensors are nonsingular. Their inverses are their transposes.