  In this paper, we introduce the tensor similar transforation and propose the
T-Jordan canonical form and its properties. The concept of T-minimal polynomial
and T-characteristic polynomial are raised. As a special case, we present
properties when two tensors commutes via the tensor T-product. The
Cayley-Hamilton theorem also holds for tensor cases. Then we focus on the
tensor decomposition theory. T-polar, T-LU, T-QR and T-Schur decomposition of
tensors are obtained. When a F-square tensor is not invertible via the
T-product, we give the T-group inverse and T-Drazin inverse which can be viewed
as the extension of matrix cases. The expression of T-group and T-Drazin
inverse are given by the T-Jordan canonical form. The polynomial form of
T-Drazin inverse is also proposed. As an application we raise the T-linear
system and get its solution. In the last part, we give the T-core-nilpotent
decomposition and show that the T-index and T-Drazin inverse can be given by a
limitation process.
