Transposes, L-Eigenvalues and Invariants of Third Order Tensors

Abstract: 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 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-eigenvalues. Here, "L" is named after Levi-Civita. 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 L-eigenvalues, singular values, C-eigenvalues and Z-eigenvalues for a third order tensor. They are all invariants of that third order tensor. For a third order partially symmetric tensor, we give its eigenvector decomposition. The piezoelectric tensor, the inverse piezoelectric tensor, and third order symmetric tensors are third order partially symmetric tensors. We make a conjecture a third order symmetric tensor has seven independent invariants. We raise the questions on how may independent invariants a third order tensor, a third order partially symmetric tensor and a third order cyclically symmetric tensor may have respectively.