Some Inverse Eigenvalue Problems For Matrices With Special Structures And Some Researches On Constrained Matrix Equations

A matrix inverse eigenvalue problem concerns the reconstruction of a matrix from prescribed spectral data. The spectral data involved may consist of the complete or only partial information of eigenvalues or eigenvectors. Inverse eigenvalue problems arise in a remarkable variety of applications. The list includes but is not limited to control design, geophysics, molecular spectroscopy, particle physics, structure analysis and so on. The constrained matrix equation problem is to find the solution of a matrix equation in a constrained matrix set. The study of it has been a hot topic in the field of numerical algebra in recent years. Actually, it is widely used in many fields such as structural design, system identification, structural dynamics, automatics control theory, vibration theory. This dissertation considers some inverse eigenvlue problems for Jacobi matrix and symmetric arrowhead matrix. Some constrained matrix equation problems are also discussed. The main works and results of this dissertation are as follows.1. The problem of constructing Jacobi matrix from its mix-type eigenpairs (the eigenpairs of the matrix itself and two of its pricinpal submatrices) is sdudied. The Jacobi matrix rank one update problems, given either their mix-type eigenpairs or their pricinpal submatrix and defective eigenpairs(the eigenvalues of the matrix and some elements of the corresponding eigenvectors) are also discussed. By analysis the structural characterizations of the eigenpairs of the Jacobi matrix, the necessary and sufficient conditions for the existence of and the general expressions for the solutions are obtained. The numerical algorithms and examples to solve the above problems are also given.2. The same with the n × n Jacobi matrix, symmetric arrowhead matrix has 2n-1 non-zero elements. Similarly to the proposing of inverse eigenvalue problems for Jacobi matrix, six kinds of inverse eigenvalue problems of symmetric arrowhead matrices are considered. The structrual characterizations of symmetric arrowhead matrices are discussed and the general expressions of solutions are obtained. Moreover, the numerical algorithms and examples are given.3. For three kinds of special matrices which have some symmetric or antisymmetric structure, the inverse problems of matrix equation AX = B on those matrices sets have been studied. In this dissertation, the Procrustes prolems of matrix equation AX = B on those matrices sets are considered. By using the subspace theory and the projection theorem, we transfer the Procrustes problem...