This talk presents two parts of results related to tensor computations. The first part is on a new matricization approach, resulting in lower and upper approximations for the CP-rank. In this part, theoretical properties of the new ranks (to be called the M-ranks) will be discussed, with applications to solve the low CP-rank tensor completion problem. The second part establishes the equivalence between a tensor and its Tucker core, in terms of the CP-rank, the Schatten norms, and the Z-eigenvalues of tensors. A new computational scheme is proposed based on the invariance properties.