Tensor ranks over the quaternions

It is well known that quaternion tensors play a vital role in some areas, such as colour image and signal processing. So, in this thesis, we mainly extend the results of tensor ranks for real and complex numbers to quaternions which is non-commutative. A tensor is defined as a multidimensional array. However, there are several ways to decompose a tensor. For example, the decomposi- tion of a tensor into the sum of rank-one tensors is often called CANDECOM, PARAFAC, or CANDECOM/PARAFAC(CP). The minimum number of rank- one tensors needed to represent a tensor is known as the rank of a tensor. It is used here to find the ranks of some quaternion tensors. Determining the tensor ranks is difficult, even for small-size tensors. Also, the tensor ranks may vary on the basis fields. Up to now, most published results on the tensor ranks are over the real and complex number fields, which are commutative. In nature, not all problems are commutative. Hence, understanding tensors and tensor ranks in non-commutative cases is also essential. Nevertheless, a limited number of papers have been published on tensor ranks in non-commutative cases. In this thesis, we describe some properties of quaternion tensors, discuss the characterization of quaternion tensor ranks, compute the tensor ranks for several quaternion tensors using some simple evaluation methods, and find the upper bounds of maximal ranks for some quaternion tensors.