Numerical Methods For Solving Some Matrix Inverse Eigenvalue Problems

In this paper,the following three kinds of inverse eigenvalue problems are condered: the generalized inverse eigenvalue problems for Hermitian and J-Hamiltonian/skew-Hamiltonian matrices,the inverse eigenvalue problems for discrete gyroscopic systems and the inverse quadratic eigenvalue problems for undamped gyroscopic systems with connectivity constraints are investigated.For the generalized inverse eigenvalue problem for Hermitian and J-Hamiltonian/skew-Hamiltonian matrices.The properties and structures of Hermitian and J-Hamiltonian/ skew-Hamiltonian matrices are analyzed.The solvability conditions and the explicit representations of the general solution for the inverse problem are given by using singular value decompositions of some matrices.For the inverse eigenvalue problems of the discrete gyroscopic systems.In the case that the partial spectral data are given,by using QR-decomposition of the modal matrix and matrix derivative,the expression of the general solution of the inverse problem is given,and then the optimal approximate solution of the known matrix pair is given.And we also give the explicit representation of the symmetric positive definite matrix and the skew-symmetric matrix when the system operates below the lowest critical speed.For the inverse quadratic eigenvalue problems for undamped gyroscopic systems with connectivity constraints.A direct method for simultaneously updating mass,gyroscopic and stiffness matrices based on incomplete modal measured data is developed,which preserves the connectivity of the original model.By using the Kronecker product and straightening functions of the matrix,the optimal approximation mass,gyroscopic and stiffness matrices which satisfy the required eigenvalue equation are obtained.