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-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.