Some properties of characteristic polynomials, eigenvalues, and eigenvectors of the Wilkinson matrices W+2n+1 and W-2n+1 are presented. It is proved that the eigenvalues of W+2n+1 just are the eigenvalues of its leadi...Some properties of characteristic polynomials, eigenvalues, and eigenvectors of the Wilkinson matrices W+2n+1 and W-2n+1 are presented. It is proved that the eigenvalues of W+2n+1 just are the eigenvalues of its leading principal submatrix Vn and a bordered matrix of Vn. Recurrence formula are given for the characteristic polynomial of W+2n+1. The eigenvectors of W+2n+1 are proved to be symmetric or skew symmetric. For W-2n+1, it is found that its eigenvalues are zero and the square roots of the eigenvalues of a bordered matrix of V2n. And the eigenvectors of W-2n+1, which the corresponding eigenvalues are opposite in pairs, have close relationship.展开更多
In this paper, we established a connection between a square matrix “A” of order “n” and a matrix defined through a new approach of the recursion relation . (where is any column matrix with n real ...In this paper, we established a connection between a square matrix “A” of order “n” and a matrix defined through a new approach of the recursion relation . (where is any column matrix with n real elements). Now the new matrix gives us a characteristic equation of matrix A and we can find the exact determination of Eigenvalues and its Eigenvectors of the matrix A. This new approach was invented by using Two eigenvector theorems along with some examples. In the subsequent paper we apply this approach by considering some examples on this invention.展开更多
The quartic minimization over the sphere is an NP-hard problem in the general case.There exist various methods for computing an approximate solution for any given instance.In practice,it is quite often that a global o...The quartic minimization over the sphere is an NP-hard problem in the general case.There exist various methods for computing an approximate solution for any given instance.In practice,it is quite often that a global optimal solution was found but without a certification.We will present in this article two classes of methods which are able to certify the global optimality,i.e.,algebraic methods and semidefinite program(SDP)relaxation methods.Several advances on these topics are summarized,accompanied with some emerged new results.We want to emphasize that for mediumor large-scaled instances,the problem is still a challenging one,due to an apparent limitation on the current force for solving SDP problems and the intrinsic one on the approximation techniques for the problem.展开更多
基金The Fundamental Research Funds for the Central Universities, China (No.10D10908)
文摘Some properties of characteristic polynomials, eigenvalues, and eigenvectors of the Wilkinson matrices W+2n+1 and W-2n+1 are presented. It is proved that the eigenvalues of W+2n+1 just are the eigenvalues of its leading principal submatrix Vn and a bordered matrix of Vn. Recurrence formula are given for the characteristic polynomial of W+2n+1. The eigenvectors of W+2n+1 are proved to be symmetric or skew symmetric. For W-2n+1, it is found that its eigenvalues are zero and the square roots of the eigenvalues of a bordered matrix of V2n. And the eigenvectors of W-2n+1, which the corresponding eigenvalues are opposite in pairs, have close relationship.
文摘In this paper, we established a connection between a square matrix “A” of order “n” and a matrix defined through a new approach of the recursion relation . (where is any column matrix with n real elements). Now the new matrix gives us a characteristic equation of matrix A and we can find the exact determination of Eigenvalues and its Eigenvectors of the matrix A. This new approach was invented by using Two eigenvector theorems along with some examples. In the subsequent paper we apply this approach by considering some examples on this invention.
基金supported by The Natural Science Foundation of Hunan Province(No.2021JJ40708)The Natural Science Foundation of the Higher Education Institutions of Jiangsu Province(No.17KJB110008)。
基金This work is partially supported by the National Natural Science Foundation of China(No.11771328)Young Elite Scientists Sponsorship Program by Tianjin,and the Natural Science Foundation of Zhejiang Province,China(No.LD19A010002).
文摘The quartic minimization over the sphere is an NP-hard problem in the general case.There exist various methods for computing an approximate solution for any given instance.In practice,it is quite often that a global optimal solution was found but without a certification.We will present in this article two classes of methods which are able to certify the global optimality,i.e.,algebraic methods and semidefinite program(SDP)relaxation methods.Several advances on these topics are summarized,accompanied with some emerged new results.We want to emphasize that for mediumor large-scaled instances,the problem is still a challenging one,due to an apparent limitation on the current force for solving SDP problems and the intrinsic one on the approximation techniques for the problem.
