Decompositions Of A Tensor Over Max Semiring

Decompositions of a tensor has wide applications in signal processing,data mining,numerical analysis and nervous system science.In this thesis,we study decompositions of a tensor over Max semiring and related problems.First,examples are given to prove that the full rank matrix can not preserve the rank of a matrix under the multiplication of matrix and matrix over Max semiring.It is proved that the generalized permutation matrix can preserve the rank of a tensor under the multiplication of matrix and tensor over Max semiring.Secondly,examples are given to prove that for some symmetric matrix there doesn't exists a symmetric decomposition.Then a kind of decomposition of symmetric tensor over Max semring is discussed in detail,and a necessary and sufficient condition for the existence of symmetric decomposition of symmetric tensor is given.Finally,the Vandermonde decomposition of Hankel tensor over Max semiring is researched.Because the Hankel tensor over Max semiring does not necessarily exist Vandermonde decomposition,a necessary and sufficient condition for the existence of the Vandermonde decomposition of Hankel tensor is given.Furthermore,a characterization of Vandermonde decomposition rank of Hankel tensor is given.