| In this thesis some nonlinear matrix problems and special structure matrix problemsare investigated.1. The undamped gyroscopic systemThe spectral decomposition theorem of undamped gyroscopic system G(λ)=Mλ~2+Cλ+K is given, where M is mass matrix, K is stiffness matrix, C is gyroscopic matrix,and the inverse eigenvalue problem as well as the spill-over eigenvalue embeddingproblem is solved by the spectral decomposition theorem. The spectral decompositiontheorem shows that once the real standard pair (X, T) is given, the spectral decompositionsof M, C, K, including a skew-symmetric parameter matrix S,are decided. If T is ablock diagonal matrix, S becomes an upper tridiagonal Hankel-like matrix. Especially,if G(λ) only has semi-simple eigenvalues, then there exists a real standard pair, such thatS only has 2n nonzero elements 1 or-1. On the other hand, the spectral decompositiontheorem presents a necessary and sufficient condition for the solvability of the quadraticinverse eigenvalue problem, and an expression of the general solution(if exists). Whenall of the eigenvalues of G(λ) are simple, the spill-over eigenvalue embedding problemis always solvable and an expression of the general solution, only including the knowneigen information, is given.2. Solvability and algebraic perturbation analysis of a class of nonlinear matrixequationThe nonlinear matrix equation X~s±A~*X~tA=Q(s＞0, t＞0) is systematicallystudied, including solvability, iterative algorithms and algebraic perturbation analysis. Atfirst, an necessary and sufficient condition and an expression of the general solution (ifexists) of X~s+A~*X~tA=Q are derived by a matrix decomposition method. Secondly,an interval in which an Hermitian positive solution exists and a sufficient condition forthe existence and uniqueness of the Hermitian positive solution are introduced by theBrouwer fixed point theorem and the eigenvalue method. For X~s-A~*X~tA=Q, we studythree cases: when s＜t, the maximal-like solution and the minimal-like solution aredefined, and sufficient conditions for their existences are given, furthermore, a sufficientcondition for the existence of the maximal solution and an iterative formula are given;when s＞t, a new iterative algorithm for the unique solution is given; when s=t, anexpression of the unique solution is obtained. Besides, the algebraic perturbation analysis theory for the unique solution, the maximal-like solution and the minimal-like solutionare established.3. Sensitivity Analysis for a class of real polynomial matrix equationThe sensitivity of the symmetric positive solution of real polynomial matrix equationX~s±A′X~tA=Q is considered. A sufficient condition of the existence of the uniquesymmetric positive solution of the perturbed equation is established. As an estimator ofthe sensitivity of the unique symmetric positive solution, a condition number is definedand its expression is derived. Numerical experiments show that the defined conditionnumber works well.4. Submatrix-constraint symmetric generalized inverse eigenvalue problem andoptimal solution problemThe descriptions in math language of the partial embedding problem and optimalsolution problem for the symmetric generalized eigenvalue model are given. A necessaryand sufficient condition for solvability and an expression of the general solution arederived.5. Generalized symmetric matrixDegree k (R, S )-symmetric matrix and (R,ω_η)-symmetric matrix are defined and theirstructures are characterized. The corresponding eigenvalue problem, singular value decomposition,least squares problem, inverse eigenvalue problem, minimal norm problemand optimal solution problem are studied. |
