Abstract
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 semidefinite) tensors, Z-tensors, M-tensors, Hankel tensors, P-tensors, B-tensors and H-tensors. Finally, as an application of permutation transformations of tensors, the 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 weakly irreducible tensors so as to handle easily.
Highlights
The study of tensors with their various applications has attracted extensive attention and interest, since the work of Qi (2005) and Lim (2005)
Some problems of higher dimension reducible matrices can be translated into the corresponding problems of lower dimension irreducible matrices by using the permutation transformation of matrices
For further discussing property of the permutation transformation of tensors, we introduce the following general product of two n-dimensional tensors defined in Shao (2013)
Summary
The study of tensors with their various applications has attracted extensive attention and interest, since the work of Qi (2005) and Lim (2005). Permutation transformations of tensors and its expression For a positive integer n, let [n] = {1, 2, . We give the definition of permutation transformation of tensors.
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