lk,s-Singular values and spectral radius of partially symmetric rectangular tensors  

作　　者：Hongmei YAO Bingsong LONG Changjiang BU Jiang ZHOU 

机构地区：[1]College of Science, Harbin Engineering University, Harbin 150001, China

出　　处：《Frontiers of Mathematics in China》2016年第3期605-622,共18页中国高等学校学术文摘·数学（英文）

基　　金：Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant Nos. 11371109, 11426075）, the Natural Science Foundation of Heilongjiang Province （No. QC2014C001）, and the Fundamental Research Funds for the Central Universities.

摘　　要：The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties oflk,singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest lk,S-singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest lk,s- singular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties oflk,singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest lk,S-singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest lk,s- singular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.

关 键 词：lk s-Singular values spectral radius positive definiteness partiallysymmetric rectangular tensor weakly irreducible 

分 类 号：O151.21[理学—数学] O183.2[理学—基础数学]