The Research On The Symmetric Rank Of The Symmetric Tensor

Symmetric tensor is a very important class of tensor.Compared to the general tensor,it has some better properties.Only the symmetric tensor has symmetric rank and the general tensor has no. As rank is portrayed as the essential characteristic of tensor,symmetric rank is the essential characteristic of symmetric tensor. So the research on the symmetric rank of the symmetric tensor is fundamental and important.Based on the introduction of the background and research status of the symmetric tensor and symmetric rank,this thesis gives the symmetric rank of a special class of symmetric tensor.The main work is as follows:Let v1 and v2 be two linearly independent vectors in C",positive integers satisfy and v2 forms a symmetric tensor according to some tensor product,denoted as S(v1(?)(k-t)(?)v2(?)t) Through this thesis,we will get:(1) S(v1(?)(k-t)(?)v2(?)t)has a minimum of symmetric outer product decomposition, ranks(S(v1(?)(k-t)(?)v2(?)t)) =t + 1, ranks(S(v1(?)(k-t1)(?)v2(?)t1+···+.v1(?)(k-tn)(?)v2(?)tn))(2) max{t1 + 1, k - tn + 1} ≤ ranks (S(v1(?)(k-t1)(?)v2(?)t1+···+v1(?)(k-tn)(?)v2(?)tn)) and ranks(S(v1(?)(k-t1)(?)v2(?)t1+···+ v1(?)(k-tn)(?)v2(?)tn)) ≤max {k-t1 +1,tn +1}.