结构矩阵多项式的实谱分解及其在模型修正中的应用

The real-valued spectral decompositions are derived for the alternating and the palindromic matrix polynomials, whose coefficient matrices are represented by the given real standard pair and a structured parameter matrix. The structure of the parameter matrix is corresponding to that of the matrix polynomial. Applying the real-valued spectral decomposition, we can get the isospectral set of the alternating and the palindromic matrix polynomials, and solve the alternating and the palindromic quadratic inverse eigenvalue problems. Especially, when both 1 and -1 are contained in the given spectrum, we present the necessary and sufficient condition of the solvability and the general solution of the palindromic quadratic inverse eigenvalue problem. At last, two model updating problems of the gyroscopic and the palindromic systems in structured dynamic systems are studied by the real-valued spectral decomposition, structure-preserving methods and other numerical methods. The necessary and sufficient conditions of the solvability and the general solutions are given. In the gyroscopic model updating problem, we study the case that the symmetry and the positive definiteness of the mass and the stiffness matrices, and the skew-symmetry of the gyroscopic matrix are need to be preserved. In these two model updating problems, we always preserve the Hamiltonian or symplectic symmetry of the spectrum, and update different types of eigenvalues by each other.

本项目首先给出交替式和回文式矩阵多项式的实谱分解, 将系数矩阵写成给定的实标准对和一个带结构的参数矩阵的表达式, 其中参数矩阵能够反映矩阵多项式的结构特征. 其次, 利用实谱分解, 构造交替式和回文式矩阵多项式的同谱类, 进而求解给定全部谱信息的交替式和回文式二次特征值反问题. 特别地, 当给定的谱信息中包含特征值1和 -1时, 利用实谱分解直接给出回文式二次特征值反问题有解的充要条件和通解表达式. 最后, 本项目完全利用实谱分解、保结构变换等数值方法, 研究两类在结构动力学中常见的陀螺系统和回文式系统的模型修正问题, 给出问题有解的充要条件和通解表达式. 在陀螺系统的模型修正问题中, 要求保持质量矩阵和刚度矩阵的对称正定性, 并且保持陀螺矩阵的反对称性.在两类模型修正过程中, 始终保持系统的特征值具有Hamiltonian对称性或辛对称性, 并且实现不同类型的特征值之间的相互更新.

本项目解决了源于结构振动力学的几类结构矩阵多项式问题和模型修正问题.给出交替式和回文式二次矩阵多项式的实谱分解，系统刻画了参数矩阵的结构.利用实谱分解解决了交替式和T-回文式二次特征值反问题和不同类型特征值之间的相互更新问题.利用一组结构矩阵作为基，刻画通解的一般表达式.完成对交替式矩阵多项式同谱类的刻画，并构造出保结构线性化.完成对陀螺系统模型修正问题的研究，并能够保持质量矩阵和刚度矩阵的正定性.建立了结构矩阵多项式的符号特征理论.