Gram方阵探讨.doc

目录 摘要……………………………………………………………………………………1 关键词…………………………………………………………………………………1 Abstract……………………………………………………………………………………1 Key words……………………………………………………………………………1 引言…………………………………………………………………………………………1 1 欧几里得空间………………………………………………………………………1 1.1 欧几里得空间……………………………………………………………………1 1.2 标准正交基………………………………………………………………………2 1.2.1 标准正交基的定义………………………………………………………………2 1.2.2 Gram-Schmidt正交化…………………………………………………………3 2 Gram方阵…………………………………………………………………………………4 2.1 Gram方阵的定义…………………………………………………………………4 2.2 Gram方阵的性质…………………………………………………………………5 3 Gram方阵的几何意义……………………………………………………………9 3.1平行六面体…………………………………………………………………9 3.1.1平行六面体的定义…………………………………………………………9 3.1.2平行六面体的体积…………………………………………………………10 3.2 Gram方阵的几何意义…………………………………………………………11 3.2.1超平行六面体…………………………………………………………11 3.2.2 Gram方阵的几何意义……………………………………………………11 4结论………………………………………………………………………………13 致谢…………………………………………………………………………………13 参考文献……………………………………………………………………………13 附录A………………………………………………………………………………14 Gram方阵的探讨 信息与计算科学 王作宾 指导老师 叶传秀 摘要：欧几里得空间是极其重要的向量空间,而Gram方阵又是欧几里得空间中的一个特殊的度量矩阵,它具有一些重要的特征.这里将对Gram方阵的定义、行列式、性质及其几何意义等做进一步探讨.首先介绍内积、欧几里得空间、标准正交基和施密特正交化等概念.然后给出在欧几里得空间中一组向量下的Gram方阵的定义及其一些重要性质,如Gram方阵的正定性、合同性等.最后探讨一下Gram方阵的几何意义,即在欧几里得空间中,Gram行列式等于超平行六面体体积的平方. 关键词: 内积 Gram方阵 标准正交基 施密特正交化 合同 超平行六面体 Discussion on Gram Matrix Information and Computing Science Wang Zuobin Tutor Ye Chuanxiu Abstract: Euclidean space is the extremely important vector space. However, Gram matrix in Euclidean space is a special measure matrix, which has some important features. Here there will be to do a further study on the definition, determinant, properties and geometry interpretation of Gram matrix. Firstly, it introduces the concepts of inner product, Euclidean space, the standard orthogonal basis and the Schmidt orthogonalization. Secondly, it presents the definition of the Gram matrix and some of its important properties under a group of vector in the Euclidean space, such as the positive definiteness and the contract of Gram matrix. Finally, it will discuss the geometry