Tensor Rank of Multi-form W Tensors

The tensor rank has been an important notion in the past century. We study the n-form W tensors in terms of their tensor rank, because such tensors have been widely realized and applied in physics theory and experiments. We begin from the W tensor with n=3 and extend it to the W tensor with n>3. For this purpose, we firstly introduce some basic knowledge such as matrices and calculation, as well as Kronecker product where we need the inner products of two product vectors or matrices. Then we introduce the notions of linearly dependent and independent vectors and matrices, to make them applicable to the evaluation of tensor rank in multi-forms. We also define the notion of tensor rank and show the difference between the tensor rank for 2-form and n-form. We develop some method such as finding the connection between the n-form and (n+1)-form tensors, so that we may employ the induction method for n.