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摘要而矩阵分解是矩阵理论中的重要组成部分，已经在各科技领域中得到了具体的应用，特别是在计算机中的广泛应用。其定义是指将一个矩阵分解成特殊矩阵（如三角矩阵）的乘积或和的形式。矩阵分解的方法有很多，其中常见的有QR分解、三角分解、满秩分解与奇异值分解等。而QR分解经常用来解线性最小二乘法问题，同时也是目前求一般矩阵全部特征值的最有效并广泛应用的方法，并且在实际应用中具有重要意义。已经成功运用在图像处理与分析、人脸识别、水印嵌入等实际应用中。本论文就矩阵的QR分解展开总结性的论述，介绍了该方法的思想、原理、步骤以及应用，并以实例详细呈现出来，并且附有用QR分解求解特征值的算法以及C语言实现。27489
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关键词 QR分解 Householder变换 Givens变换 Schmidt正交化 Hessenberg矩阵 特征值 最小二乘法
毕业论文设计说明书外文摘要
Title QR decomposition of the matrix and its application
Abstract
The matrix decomposition is an integral part of matrix theory, has been a particular application in the fields of technology, especially in the computer widely used. Which is defined as a matrix decomposition into a special matrix (such as triangular matrix) and the product or form. There matrix decomposition method, many of them common with QR decomposition, triangular decomposition, full rank decomposition and singular value decomposition and the like. The QR decomposition is often used to solve the linear least squares problem, is also currently seeking the most effective general all eigenvalues of matrix and widely used method, and has great significance in practical applications. It has been successfully used in image processing and analysis, face recognition, watermarking and other practical applications. The present paper on matrix QR decomposition expanded summary exposition introduces the ideas, principles, procedures, and use of the method, and examples presented in detail and accompanied with a QR decomposition algorithm for solving eigenvalues and C language.
Keywords QR decomposition Householder Transformation Givens Transformation
Eigenvalues Linear least square method
1 绪论 1
1.1 研究背景及现状 1
1.2 线性代数的一些基本概念 1
1.2.1 范数、线性赋范空间 1
1.2.2 Hessenberg矩阵 2
1.2.3 正定矩阵 2
1.3 Householder变换 2
1.3.1 Householder变换定义 2
1.3.2 Householder变换性质 2
1.4 Givens变换 4
1.4.1 Givens变换定义 4
1.4.2 Givens变换性质 4
2 QR分解 6
2.1 QR分解定义 6
2.2 QR定理及QR分解方法 6
2.3 分解唯一性定理 8
3 QR算法求特征值 11
3.1 QR算法基本思想 11
3.2 QR算法的收敛性 12
3.3 QR算法迭代过程 13
3.4 QR算法迭代控制 20
3.5 带原点位移的QR算法 21
3.6 QR分解求特征值算法源程序 21
3.6.1 各函数功能实现 21
3.6.2 数据实验 22