  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.
