| 刘岩,南基洙.有限局部环上对称矩阵的计数定理(I)[J].数学研究及应用,2006,26(3):423~439 |
| 有限局部环上对称矩阵的计数定理(I) |
| Anzahl Theorems in Symmetric Matrices over Finite Local Rings (I) |
| 投稿时间：2004-06-11 |
| DOI：10.3770/j.issn:1000-341X.2006.03.002 |
| 中文关键词: 合同变换 对称矩阵标准型 正交群 轨道. |
| 英文关键词:congruent transformation normal form of symmetric matrix orthogonal group. |
| 基金项目:the Key Project of Chinese Ministry of Education (03060) |
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| 摘要点击次数: 3393 |
| 全文下载次数: 2819 |
| 中文摘要: |
| 设$A_{n}(R)$是有限局部环$Z/p^{k}Z$上$n$阶对称矩阵的集合, 这里$n\geq 2$. $p$是大于$2$素数, $p\equiv1({\rm mod}4)$ 且$k>1$. 通过确定有限局部环$Z/p^{k}Z$上对称矩阵的标准型, 计算出$A_{n}(R)$在线性群${\rm GL}_{n}(R)$作用下的轨道数, 从而计算出由特定对称矩阵确定的正交群的阶以及与特定对称矩阵在同一轨道的对称矩阵的阶. |
| 英文摘要: |
| Let $A_{n}(R)$ be the set of symmetric matrices over $Z/p^{k}Z$ with order $n$, where $n\geq 2$, $p$ is a prime, $p>2$ and $p\equiv1({\rm mod}4)$, $k>1$. By determining the normal form of $n$ by $n$ symmetric matrices over $Z/p^{k}Z$, we compute the number of the orbits of $ A_{n}(R)$ and then compute the order of the orthogonal group determined by the special symmetric matrix. Finally we get the number of the symmetric matrices which are in the same orbit with the special symmetric matrix. |
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