二阶系统广泛产生于控制、振动等工程实际领域，在对其进行结构动力特性分析时，便会涉及到二阶系统的解耦问题，即寻找主坐标变换将三个系数矩阵同时对角化，其实现条件苛刻。虽然，数值代数领域基于Lancaster结构研究二阶系统解耦已取得一定成果，但相关算法的保结构和保谱等性质的实现仍需进一步完善提高。本项目针对基于Lancaster结构二阶系统解耦方法中的保持问题进行深入研究，以求进一步完善已有解耦算法，并为其在工程实践中应用奠定基础。首先，研究解耦变换的有效数值求解算法及解耦变换的保对称性。解耦变换描述了系统解耦前后的一种非线性关系，研究其求解及性质刻画对工程应用具有重要意义；其次，研究解耦变换的保结构和保谱特性的刻划与实现。要使解耦有意义就必须保谱，而要保谱就必须保结构，因此，研究解耦变换的这两个保持性质至关重要；最后，研究一类齐次Sylvester方程非奇异解的存在、求解算法及解的保对称性。

Quadratic systems arise extensively in many applications, such as control systems. In structural dynamic characteristics analysis of the dynamic system, Quadratic systems decoupling will be involved. It means to find coordinate transformations diagnalizing three coefficient matrices simultaneous, which is very harsh to accomplish. Although it had achieved some results that researching the problem based on Lancaster structure in numerical algebra, the implemention of structure preserving and isospectral characteristcs of relative algorithm remains to perfect further.. Focuing on the preserver problems researchees in quadratic system decoupling based on Lancaster structrue, this project is dedicated to improve and perfect the existing agorithm, and lay the foundation for it''s application in engineering practices. Firstly, effective numerical algorithms is researched to find decoupling transformations, along with its symmetric preserving characteristic. Decoupling transformations describe a kind of nonlinear of relationships between systems before and after decoupling, which is significant for engineering application. Secondly, the structure and spectral preserving problems are considered and descripted. Isospectrality is the key of decoupling, furthermore, Lancaster structure preserving is the key of isospectrality, thus, it is extremely important to reserch these two preserver problems. Finally, the existence and symetric characteristics of nonsingular solution of a kind of homogeneous Sylvester equation are researched.