The Jacobi Matrix With The Centre And The Anti-center Symmetric Matrix Inverse Eigenvalue Problem

There are all kinds of inverse eigenvalue and generalized inverse eigenvalue problems in the fields of structural design, vibration system, automation control and matrix decision etc. A compound inverse eigenvalue problem of Jacobi matrix and some generalized inverse eigenvalue problems have been discussed in this paper.From special to normal, the problem of constructing a Jacobi matrix with a prescribed principle submatrix and two ordered defective eigen-pairs has been solved gradually in this paper. First we discuss the case that the defective eigenvects are the maximum and the minimal; then the maximum and the next maximum, the maximum and the general; and the case that all are general in the last. Based on the relationship between the eigenvalue order and the number of variations of the corresponding eigenvector , some necessary and sufficient conditions for the solvability of the above problems have been derived, and the corresponding algorithms are developed at the same time. Numerical experiments show that these algorithms are quite reliable and efficient.Based on the foundational properties of centrosymmetric matrix and anti-centrosymmetric matrix, employing the technology of lowing dimension, the generalized inverse eigenvalue problems which involve centrosymmetric matrix, anti-centrosymmetric matrix, and compound centrosymmetric matrix and anti-centrosymmetric matrix, have been discussed. On the basis of above, and employing projection theory, the optimal approximation of them under spectral constraints has been solved. The formulas of general solution have been given, as well as the formulas of optimal approximation solution and numerical algorithms...