Strictly nonnegative tensors and nonnegative tensor partition  

作　　者：HU ShengLong HUANG ZhengHai QI LiQun 

机构地区：[1]Department of Applied Mathematics,The Hong Kong Polytechnic University [2]Department of Mathematics,School of Science,Tianjin University [3]The Center for Applied Mathematics of Tianjin University [4]Tianjin University

出　　处：《Science China Mathematics》2014年第1期181-195,共15页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China (Grant No. 11171252);the Hong Kong Research Grant Council (Grant Nos. PolyU 501909,502510,502111 and 501212)

摘　　要：We introduce a new class of nonnegative tensors—strictly nonnegative tensors.A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa.We show that the spectral radius of a strictly nonnegative tensor is always positive.We give some necessary and su?cient conditions for the six wellconditional classes of nonnegative tensors,introduced in the literature,and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors.We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility.We show that for a nonnegative tensor T,there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible;and the spectral radius of T can be obtained from those spectral radii of the induced tensors.In this way,we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption.Some preliminary numerical results show the feasibility and effectiveness of the algorithm.We introduce a new class of nonnegative tensors—strictly nonnegative tensors.A weakly irreducible nonnegative tensor is a strictly nonnegative tensor but not vice versa.We show that the spectral radius of a strictly nonnegative tensor is always positive.We give some necessary and su？cient conditions for the six wellconditional classes of nonnegative tensors,introduced in the literature,and a full relationship picture about strictly nonnegative tensors with these six classes of nonnegative tensors.We then establish global R-linear convergence of a power method for finding the spectral radius of a nonnegative tensor under the condition of weak irreducibility.We show that for a nonnegative tensor T,there always exists a partition of the index set such that every tensor induced by the partition is weakly irreducible;and the spectral radius of T can be obtained from those spectral radii of the induced tensors.In this way,we develop a convergent algorithm for finding the spectral radius of a general nonnegative tensor without any additional assumption.Some preliminary numerical results show the feasibility and effectiveness of the algorithm.

关 键 词：NONNEGATIVE tensor spectral radius strict nonnegativity weak IRREDUCIBILITY 

分 类 号：O183.2[理学—数学]