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 rea...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.展开更多
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 study the singular values/vectors problem of real ...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 study the singular values/vectors problem of real nonnegative partially symmetric rectangular tensors. We first introduce the concepts of/k,S-singular values/vectors of real partially symmetric rectangular tensors. Then, based upon the presented properties of lk,S-singular values /vectors, some properties of the related /k＇S-spectral radius are discussed. Furthermore, we prove two analogs of Perron-Frobenius theorem and weak Perron-Frobenius theorem for real nonnegative partially symmetric rectangular tensors.展开更多
By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial....By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial. The definition of the regularity of square tensors is generalized to the rectangular tensors, and in the regular case, a root of the Echaracteristic polynomial of a special rectangular tensor is an E-singular value of the rectangular tensor. Moreover, the best rank-one approximation of a real partially symmetric rectangular tensor is investigated.展开更多
We give a Brualdi-type Z-eigenvalue inclusion set of tensors,and prove that it is tighter than the inclusion set given by G.Wang,G.L.Zhou,and L.Caccetta[Discrete Contin.Dyn.Syst.Ser.B,2017,22:187–198]in a special cas...We give a Brualdi-type Z-eigenvalue inclusion set of tensors,and prove that it is tighter than the inclusion set given by G.Wang,G.L.Zhou,and L.Caccetta[Discrete Contin.Dyn.Syst.Ser.B,2017,22:187–198]in a special case.We also give an inclusion set for l^k,s-singular values of rectangular tensors.展开更多
An algorithm for finding the largest singular value of a nonnegative rectangular tensor was recently proposed by Chang, Qi, and Zhou [J. Math. Anal. Appl., 2010, 370： 284-294]. In this paper, we establish a linear co...An algorithm for finding the largest singular value of a nonnegative rectangular tensor was recently proposed by Chang, Qi, and Zhou [J. Math. Anal. Appl., 2010, 370： 284-294]. In this paper, we establish a linear conver- gence rate of the Chang-Qi-Zhou algorithm under a reasonable assumption.展开更多
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.
文摘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.
文摘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 study the singular values/vectors problem of real nonnegative partially symmetric rectangular tensors. We first introduce the concepts of/k,S-singular values/vectors of real partially symmetric rectangular tensors. Then, based upon the presented properties of lk,S-singular values /vectors, some properties of the related /k＇S-spectral radius are discussed. Furthermore, we prove two analogs of Perron-Frobenius theorem and weak Perron-Frobenius theorem for real nonnegative partially symmetric rectangular tensors.
文摘By the resultant theory, the E-characteristic polynomial of a real rectangular tensor is defined. It is proved that an E-singular value of a real rectangular tensor is always a root of the E-characteristic polynomial. The definition of the regularity of square tensors is generalized to the rectangular tensors, and in the regular case, a root of the Echaracteristic polynomial of a special rectangular tensor is an E-singular value of the rectangular tensor. Moreover, the best rank-one approximation of a real partially symmetric rectangular tensor is investigated.
基金This work was supported in part by the National Natural Science Foundation of China(Grant No.11801115)the Youth Science Foundation of Heilongjiang Province of China(No.QC2018002)the Fundamental Research Funds for Central Universities.
文摘We give a Brualdi-type Z-eigenvalue inclusion set of tensors,and prove that it is tighter than the inclusion set given by G.Wang,G.L.Zhou,and L.Caccetta[Discrete Contin.Dyn.Syst.Ser.B,2017,22:187–198]in a special case.We also give an inclusion set for l^k,s-singular values of rectangular tensors.
文摘An algorithm for finding the largest singular value of a nonnegative rectangular tensor was recently proposed by Chang, Qi, and Zhou [J. Math. Anal. Appl., 2010, 370： 284-294]. In this paper, we establish a linear conver- gence rate of the Chang-Qi-Zhou algorithm under a reasonable assumption.
