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
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.
