Fourth-order cartesian tensors: Old and new facts, notions and applications

Fourth-order tensors can be represented in many different ways. For instance, they can be represented as multilinear maps or multilinear forms. It is also possible to describe a fourth-order tensor in a given vector space by a second-order tensor but in another vector space with higher dimension. Such a representation makes the manipulation of fourth-order tensors similar to that of the more familiar second-order tensors. In this paper, we use these three descriptions to discuss the different symmetries of fourth-order tensors, to present the algebra of the space of fourth-order symmetric tensors and to describe different metrics on this space. Isotropic tensors and orientation tensors are presented using these different representations of fourth-order tensors. Applications to elasticity and high angular resolution diffusion imaging are discussed. Finally, we present a systematic and consistent approach to finding the tensor of an even order that best fits in some sense a tensor of higher even order.