Permutation Transformations Of Tensors With Applications

The permutation transformation of matrices plays a very important role in linear algebra and matrix theory. Inspired by this, we studied the permutation transformations of tensors in this thesis.The permutation transformation of tensors is introduced and its basic properties are discussed. The invariance under permutation transformations is studied for some important structure tensors such as symmetric tensors, positive definite (positive semi-definite) tensors, Z-tensors, M-tensors, P-tensors, B-tensors and H-tensors. In addition, as an application of permutation transformations of tensors, an equivalence definition of reducible tensors is given by using the permutation transformation of tensors, and the reducible canonical form theorem of tensors is given. The theorem shows that some problems of higher dimension tensors can be translated into the corresponding problems of lower dimension irreducible tensors so as to handle easily.