Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence
Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence
复制标题
同余下的辛空间以及对称和非奇异斜对称矩阵对
DOI：
10.1016/j.laa.2017.09.026
复制
发表时间：
2017
影响因子：
1.1
通讯作者：
V. Sergeichuk
中科院分区：
文献类型：
--
作者：
V. Bovdi;R. Horn;M. Salim;V. Sergeichuk
设F是特征不为2的域，（A，B）是F上的一对n× n矩阵，其中A是对称的，B是反对称的. Sergeichuk（1988）[25]给出了（A，B）关于同余变换（S T A S，S T B S）的标准型，直到F的有限扩张上的对称型和Hermitian型的分类。当B非奇异时，得到了（A，B）的一个较简单的标准型.这样的对（A，B）定义了辛空间上的二次型，也就是说，定义了向量空间上的二次型，其标积由非奇异反对称形式给出。作为应用，我们得到了真实的和复辛空间上二次型和Hamilton算子的标准矩阵。
Let F be a field of characteristic not 2, and let (A, B) be a pair of n× n matrices over F, in which A is symmetric and B is skew-symmetric. A canonical form of (A, B) with respect to congruence transformations (S T A S, S T B S) was given by Sergeichuk (1988)[25] up to classification of symmetric and Hermitian forms over finite extensions of F. We obtain a simpler canonical form of (A, B) if B is nonsingular. Such a pair (A, B) defines a quadratic form on a symplectic space, that is, on a vector space with scalar product given by a nonsingular skew-symmetric form. As an application, we obtain known canonical matrices of quadratic forms and Hamiltonian operators on real and complex symplectic spaces.