正交变换的等价条件及其应用

目录

1引言 ................................................................................................................... １2正交变换的定义及其等价条件 ........................................................................ １

2.1定义..................................................................................................................................１

2.2等价条件..........................................................................................................................２3正交变换的应用................................................................................................ ４

3.1化二次型为标准形..........................................................................................................４

3.2解不变子空间相关问题..................................................................................................８

3.3求解矩阵问题..................................................................................................................８

3.4求解欧氏空间中其它相关问题......................................................................................８

3.5在积分中的应用.......................................................................................................... １１4结束语 ............................................................................................................ １２参考文献 ........................................................................................................... １３致谢语 ............................................................................................................... １４

正交变换的等价条件及其应用

数学系2013级1班许鹏

指导教师：陈金梅

摘要:正交变换在大学学习中是一个重要的概念,例如在代数中,它涉及到了线性代数中一大部分的基本概念,如矩阵、向量、线性变换、标准正交基等,深入探讨研究这个课题对学好高等代数和线性代数十分有帮助.不仅如此,它在其他的领域也有着大范围的普及,如在积分的应用中,在多重积分的方面。本文首先叙述了正交变换的最基础的概念,从它的定义开始,探究它在代数书中的一些特点和求解过程,主要就正交变换进行探索研究,得出它的几种等价条件,为了体现它的重要作用,我们将做一些例证,举例说明它的价值。

关键词:正交变换;标准正交基;内积;正交矩阵。

Equivalent Conditions of Orthogonal Transformation and Their

Applications

Xupeng

Class 1, Mathematics Department

Tutor:ChenJinMei

Abstract: Orthogonal transformation is an important concept in university learning. For example, in algebra, it involves a basic concept of a large part of linear algebra, such as matrix, vector, linear transformation, standard orthogonal basis, and so on. It is very helpful to learn higher algebra and linear algebra, and it is also popular in other fields, such as in the application of integral points, in the case of multiple points. This paper first describes the most basic concept of orthogonal transformation, from its definition, to explore its characteristics in the algebra of the book and the process of solving the main orthogonal transformation to explore the study, to obtain several of its equivalent conditions , In order to reflect its important role, we will do some examples, exemplify its value.

Key words:Orthogonal transformation; standard orthogonal basis; inner product; orthogonal matrix.