Inverse Problem Of Quadratic Eigenvalue Of Matrix And Its Application

Inverse quadratic eigenvalue problem of matrices is widely used in many fields and is closely related to practical problems.The research on inverse quadratic eigenvalue problem promotes the development of other disciplines and has high scientific research value.For quadratic inverse eigenvalue problems,one solution is to reduce the original second order inverse problems to the first order inverse problem.However,this method will destroy the special structure of the matrix in the original problem.In this thesis,the inverse quadratic eigenvalue problem is studied by matrix decomposition,the best approximation solution is analyzed and the coefficient matrix of the quadratic equation is corrected.The research contents are as follows:1.The quadratic matrix equation MX?~2+CX?+KX=0 is studied under the condition of given partial eigenvalues and eigenvectors,using matrix block method?singular value decomposition and Moore-Penrose's inverse theory is used to discuss the existence of Hermitian R-symmetric solution,and the general expression of Hermitian R-symmetric solution is obtained,and its best approximation is obtained.Finally,an example is given to verify the correctness of the results and the effectiveness of the algorithm by using Matlab mathematical programming language.2.Furthermore,the existence of Hermitian R-antisymmetric solution of quadratic matrix equation MX?~2+CX?-KX=0 is studied in the complex field,the general expression of Hermitian R-antisymmetric solution is obtained,and its best approxima-tion is obtained.An example is given to verify the correctness of the results and the effectiveness of the algorithm.3.A updating problem of undamped gyroscopic systems is investigated.The inverse quadratic eigenvalue problem of the coefficient matrix K under the perturbation of bisymmetric matrix ?K is discussed for the matrix equation MX?~2+GX?+(?K+K)X=0.The sufficient and necessary conditions for the existence of solutions and the general expressions of solutions are given by using the structural characteristics of bisymmetric matrices and matrix decomposition theory.